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Theorem yonedainv 18272
Description: The Yoneda Lemma with explicit inverse. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y π‘Œ = (Yonβ€˜πΆ)
yoneda.b 𝐡 = (Baseβ€˜πΆ)
yoneda.1 1 = (Idβ€˜πΆ)
yoneda.o 𝑂 = (oppCatβ€˜πΆ)
yoneda.s 𝑆 = (SetCatβ€˜π‘ˆ)
yoneda.t 𝑇 = (SetCatβ€˜π‘‰)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomFβ€˜π‘„)
yoneda.r 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (πœ‘ β†’ 𝐢 ∈ Cat)
yoneda.w (πœ‘ β†’ 𝑉 ∈ π‘Š)
yoneda.u (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
yoneda.v (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
yoneda.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))))
yonedainv.i 𝐼 = (Invβ€˜π‘…)
yonedainv.n 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)))))
Assertion
Ref Expression
yonedainv (πœ‘ β†’ 𝑀(𝑍𝐼𝐸)𝑁)
Distinct variable groups:   𝑓,π‘Ž,𝑔,π‘₯,𝑦, 1   𝑒,π‘Ž,𝑔,𝑦,𝐢,𝑓,π‘₯   𝐸,π‘Ž,𝑓,𝑔,𝑒,𝑦   𝐡,π‘Ž,𝑓,𝑔,𝑒,π‘₯,𝑦   𝑁,π‘Ž   𝑂,π‘Ž,𝑓,𝑔,𝑒,π‘₯,𝑦   𝑆,π‘Ž,𝑓,𝑔,𝑒,π‘₯,𝑦   𝑔,𝑀,𝑒,𝑦   𝑄,π‘Ž,𝑓,𝑔,𝑒,π‘₯   𝑇,𝑓,𝑔,𝑒,𝑦   πœ‘,π‘Ž,𝑓,𝑔,𝑒,π‘₯,𝑦   𝑒,𝑅   π‘Œ,π‘Ž,𝑓,𝑔,𝑒,π‘₯,𝑦   𝑍,π‘Ž,𝑓,𝑔,𝑒,π‘₯,𝑦
Allowed substitution hints:   𝑄(𝑦)   𝑅(π‘₯,𝑦,𝑓,𝑔,π‘Ž)   𝑇(π‘₯,π‘Ž)   π‘ˆ(π‘₯,𝑦,𝑒,𝑓,𝑔,π‘Ž)   1 (𝑒)   𝐸(π‘₯)   𝐻(π‘₯,𝑦,𝑒,𝑓,𝑔,π‘Ž)   𝐼(π‘₯,𝑦,𝑒,𝑓,𝑔,π‘Ž)   𝑀(π‘₯,𝑓,π‘Ž)   𝑁(π‘₯,𝑦,𝑒,𝑓,𝑔)   𝑉(π‘₯,𝑦,𝑒,𝑓,𝑔,π‘Ž)   π‘Š(π‘₯,𝑦,𝑒,𝑓,𝑔,π‘Ž)

Proof of Theorem yonedainv
Dummy variables 𝑏 β„Ž π‘˜ 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.r . . 3 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
2 eqid 2725 . . . 4 (𝑄 Γ—c 𝑂) = (𝑄 Γ—c 𝑂)
3 yoneda.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
43fucbas 17950 . . . 4 (𝑂 Func 𝑆) = (Baseβ€˜π‘„)
5 yoneda.o . . . . 5 𝑂 = (oppCatβ€˜πΆ)
6 yoneda.b . . . . 5 𝐡 = (Baseβ€˜πΆ)
75, 6oppcbas 17698 . . . 4 𝐡 = (Baseβ€˜π‘‚)
82, 4, 7xpcbas 18168 . . 3 ((𝑂 Func 𝑆) Γ— 𝐡) = (Baseβ€˜(𝑄 Γ—c 𝑂))
9 eqid 2725 . . 3 ((𝑄 Γ—c 𝑂) Nat 𝑇) = ((𝑄 Γ—c 𝑂) Nat 𝑇)
10 yoneda.y . . . . 5 π‘Œ = (Yonβ€˜πΆ)
11 yoneda.1 . . . . 5 1 = (Idβ€˜πΆ)
12 yoneda.s . . . . 5 𝑆 = (SetCatβ€˜π‘ˆ)
13 yoneda.t . . . . 5 𝑇 = (SetCatβ€˜π‘‰)
14 yoneda.h . . . . 5 𝐻 = (HomFβ€˜π‘„)
15 yoneda.e . . . . 5 𝐸 = (𝑂 evalF 𝑆)
16 yoneda.z . . . . 5 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17 yoneda.c . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
18 yoneda.w . . . . 5 (πœ‘ β†’ 𝑉 ∈ π‘Š)
19 yoneda.u . . . . 5 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
20 yoneda.v . . . . 5 (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
2110, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 17, 18, 19, 20yonedalem1 18263 . . . 4 (πœ‘ β†’ (𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)))
2221simpld 493 . . 3 (πœ‘ β†’ 𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
2321simprd 494 . . 3 (πœ‘ β†’ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
24 yonedainv.i . . 3 𝐼 = (Invβ€˜π‘…)
25 eqid 2725 . . 3 (Invβ€˜π‘‡) = (Invβ€˜π‘‡)
26 yoneda.m . . . 4 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))))
2710, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 17, 18, 19, 20, 26yonedalem3 18271 . . 3 (πœ‘ β†’ 𝑀 ∈ (𝑍((𝑄 Γ—c 𝑂) Nat 𝑇)𝐸))
2817adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ 𝐢 ∈ Cat)
2918adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ 𝑉 ∈ π‘Š)
3019adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
3120adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
32 simprl 769 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ β„Ž ∈ (𝑂 Func 𝑆))
33 simprr 771 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ 𝑀 ∈ 𝐡)
3410, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 28, 29, 30, 31, 32, 33, 26yonedalem3a 18265 . . . . . . . . . 10 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘€π‘€) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€))) ∧ (β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢(β„Ž(1st β€˜πΈ)𝑀)))
3534simprd 494 . . . . . . . . 9 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢(β„Ž(1st β€˜πΈ)𝑀))
3628adantr 479 . . . . . . . . . . . 12 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ 𝐢 ∈ Cat)
3729adantr 479 . . . . . . . . . . . 12 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ 𝑉 ∈ π‘Š)
3830adantr 479 . . . . . . . . . . . 12 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
3931adantr 479 . . . . . . . . . . . 12 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
40 simplrl 775 . . . . . . . . . . . 12 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ β„Ž ∈ (𝑂 Func 𝑆))
41 simplrr 776 . . . . . . . . . . . 12 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ 𝑀 ∈ 𝐡)
42 yonedainv.n . . . . . . . . . . . 12 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), π‘₯ ∈ 𝐡 ↦ (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)))))
43 simpr 483 . . . . . . . . . . . 12 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ 𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€))
4410, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 36, 37, 38, 39, 40, 41, 42, 43yonedalem4c 18268 . . . . . . . . . . 11 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((β„Žπ‘π‘€)β€˜π‘) ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž))
4544fmpttd 7120 . . . . . . . . . 10 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ ((β„Žπ‘π‘€)β€˜π‘)):((1st β€˜β„Ž)β€˜π‘€)⟢(((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž))
466fvexi 6906 . . . . . . . . . . . . . . 15 𝐡 ∈ V
4746mptex 7231 . . . . . . . . . . . . . 14 (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’))) ∈ V
48 eqid 2725 . . . . . . . . . . . . . 14 (𝑒 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’)))) = (𝑒 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’))))
4947, 48fnmpti 6693 . . . . . . . . . . . . 13 (𝑒 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’)))) Fn ((1st β€˜β„Ž)β€˜π‘€)
50 simpl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓 = β„Ž ∧ π‘₯ = 𝑀) β†’ 𝑓 = β„Ž)
5150fveq2d 6896 . . . . . . . . . . . . . . . . . 18 ((𝑓 = β„Ž ∧ π‘₯ = 𝑀) β†’ (1st β€˜π‘“) = (1st β€˜β„Ž))
52 simpr 483 . . . . . . . . . . . . . . . . . 18 ((𝑓 = β„Ž ∧ π‘₯ = 𝑀) β†’ π‘₯ = 𝑀)
5351, 52fveq12d 6899 . . . . . . . . . . . . . . . . 17 ((𝑓 = β„Ž ∧ π‘₯ = 𝑀) β†’ ((1st β€˜π‘“)β€˜π‘₯) = ((1st β€˜β„Ž)β€˜π‘€))
54 simplr 767 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 = β„Ž ∧ π‘₯ = 𝑀) ∧ 𝑦 ∈ 𝐡) β†’ π‘₯ = 𝑀)
5554oveq2d 7432 . . . . . . . . . . . . . . . . . . 19 (((𝑓 = β„Ž ∧ π‘₯ = 𝑀) ∧ 𝑦 ∈ 𝐡) β†’ (𝑦(Hom β€˜πΆ)π‘₯) = (𝑦(Hom β€˜πΆ)𝑀))
56 simpll 765 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 = β„Ž ∧ π‘₯ = 𝑀) ∧ 𝑦 ∈ 𝐡) β†’ 𝑓 = β„Ž)
5756fveq2d 6896 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 = β„Ž ∧ π‘₯ = 𝑀) ∧ 𝑦 ∈ 𝐡) β†’ (2nd β€˜π‘“) = (2nd β€˜β„Ž))
58 eqidd 2726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 = β„Ž ∧ π‘₯ = 𝑀) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 = 𝑦)
5957, 54, 58oveq123d 7437 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓 = β„Ž ∧ π‘₯ = 𝑀) ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯(2nd β€˜π‘“)𝑦) = (𝑀(2nd β€˜β„Ž)𝑦))
6059fveq1d 6894 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 = β„Ž ∧ π‘₯ = 𝑀) ∧ 𝑦 ∈ 𝐡) β†’ ((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”) = ((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”))
6160fveq1d 6894 . . . . . . . . . . . . . . . . . . 19 (((𝑓 = β„Ž ∧ π‘₯ = 𝑀) ∧ 𝑦 ∈ 𝐡) β†’ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’) = (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’))
6255, 61mpteq12dv 5234 . . . . . . . . . . . . . . . . . 18 (((𝑓 = β„Ž ∧ π‘₯ = 𝑀) ∧ 𝑦 ∈ 𝐡) β†’ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)) = (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’)))
6362mpteq2dva 5243 . . . . . . . . . . . . . . . . 17 ((𝑓 = β„Ž ∧ π‘₯ = 𝑀) β†’ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’))) = (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’))))
6453, 63mpteq12dv 5234 . . . . . . . . . . . . . . . 16 ((𝑓 = β„Ž ∧ π‘₯ = 𝑀) β†’ (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)))) = (𝑒 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’)))))
65 fvex 6905 . . . . . . . . . . . . . . . . 17 ((1st β€˜β„Ž)β€˜π‘€) ∈ V
6665mptex 7231 . . . . . . . . . . . . . . . 16 (𝑒 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’)))) ∈ V
6764, 42, 66ovmpoa 7573 . . . . . . . . . . . . . . 15 ((β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡) β†’ (β„Žπ‘π‘€) = (𝑒 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’)))))
6867adantl 480 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Žπ‘π‘€) = (𝑒 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’)))))
6968fneq1d 6642 . . . . . . . . . . . . 13 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘π‘€) Fn ((1st β€˜β„Ž)β€˜π‘€) ↔ (𝑒 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)𝑀) ↦ (((𝑀(2nd β€˜β„Ž)𝑦)β€˜π‘”)β€˜π‘’)))) Fn ((1st β€˜β„Ž)β€˜π‘€)))
7049, 69mpbiri 257 . . . . . . . . . . . 12 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Žπ‘π‘€) Fn ((1st β€˜β„Ž)β€˜π‘€))
71 dffn5 6952 . . . . . . . . . . . 12 ((β„Žπ‘π‘€) Fn ((1st β€˜β„Ž)β€˜π‘€) ↔ (β„Žπ‘π‘€) = (𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ ((β„Žπ‘π‘€)β€˜π‘)))
7270, 71sylib 217 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Žπ‘π‘€) = (𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ ((β„Žπ‘π‘€)β€˜π‘)))
735oppccat 17703 . . . . . . . . . . . . . 14 (𝐢 ∈ Cat β†’ 𝑂 ∈ Cat)
7417, 73syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑂 ∈ Cat)
7574adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ 𝑂 ∈ Cat)
7620unssbd 4182 . . . . . . . . . . . . . . 15 (πœ‘ β†’ π‘ˆ βŠ† 𝑉)
7718, 76ssexd 5319 . . . . . . . . . . . . . 14 (πœ‘ β†’ π‘ˆ ∈ V)
7812setccat 18073 . . . . . . . . . . . . . 14 (π‘ˆ ∈ V β†’ 𝑆 ∈ Cat)
7977, 78syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑆 ∈ Cat)
8079adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ 𝑆 ∈ Cat)
8115, 75, 80, 7, 32, 33evlf1 18211 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Ž(1st β€˜πΈ)𝑀) = ((1st β€˜β„Ž)β€˜π‘€))
8210, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 28, 29, 30, 31, 32, 33yonedalem21 18264 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Ž(1st β€˜π‘)𝑀) = (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž))
8372, 81, 82feq123d 6706 . . . . . . . . . 10 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘π‘€):(β„Ž(1st β€˜πΈ)𝑀)⟢(β„Ž(1st β€˜π‘)𝑀) ↔ (𝑏 ∈ ((1st β€˜β„Ž)β€˜π‘€) ↦ ((β„Žπ‘π‘€)β€˜π‘)):((1st β€˜β„Ž)β€˜π‘€)⟢(((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž)))
8445, 83mpbird 256 . . . . . . . . 9 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Žπ‘π‘€):(β„Ž(1st β€˜πΈ)𝑀)⟢(β„Ž(1st β€˜π‘)𝑀))
85 fcompt 7138 . . . . . . . . . . 11 (((β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢(β„Ž(1st β€˜πΈ)𝑀) ∧ (β„Žπ‘π‘€):(β„Ž(1st β€˜πΈ)𝑀)⟢(β„Ž(1st β€˜π‘)𝑀)) β†’ ((β„Žπ‘€π‘€) ∘ (β„Žπ‘π‘€)) = (π‘˜ ∈ (β„Ž(1st β€˜πΈ)𝑀) ↦ ((β„Žπ‘€π‘€)β€˜((β„Žπ‘π‘€)β€˜π‘˜))))
8635, 84, 85syl2anc 582 . . . . . . . . . 10 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘€π‘€) ∘ (β„Žπ‘π‘€)) = (π‘˜ ∈ (β„Ž(1st β€˜πΈ)𝑀) ↦ ((β„Žπ‘€π‘€)β€˜((β„Žπ‘π‘€)β€˜π‘˜))))
8781eleq2d 2811 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (π‘˜ ∈ (β„Ž(1st β€˜πΈ)𝑀) ↔ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)))
8887biimpa 475 . . . . . . . . . . . . 13 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ (β„Ž(1st β€˜πΈ)𝑀)) β†’ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€))
8928adantr 479 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ 𝐢 ∈ Cat)
9029adantr 479 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ 𝑉 ∈ π‘Š)
9130adantr 479 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
9231adantr 479 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
93 simplrl 775 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ β„Ž ∈ (𝑂 Func 𝑆))
94 simplrr 776 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ 𝑀 ∈ 𝐡)
9510, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 89, 90, 91, 92, 93, 94, 26yonedalem3a 18265 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((β„Žπ‘€π‘€) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€))) ∧ (β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢(β„Ž(1st β€˜πΈ)𝑀)))
9695simpld 493 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ (β„Žπ‘€π‘€) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€))))
9796fveq1d 6894 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((β„Žπ‘€π‘€)β€˜((β„Žπ‘π‘€)β€˜π‘˜)) = ((π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€)))β€˜((β„Žπ‘π‘€)β€˜π‘˜)))
9872, 44fmpt3d 7121 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Žπ‘π‘€):((1st β€˜β„Ž)β€˜π‘€)⟢(((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž))
9998ffvelcdmda 7089 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((β„Žπ‘π‘€)β€˜π‘˜) ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž))
100 fveq1 6891 . . . . . . . . . . . . . . . . 17 (π‘Ž = ((β„Žπ‘π‘€)β€˜π‘˜) β†’ (π‘Žβ€˜π‘€) = (((β„Žπ‘π‘€)β€˜π‘˜)β€˜π‘€))
101100fveq1d 6894 . . . . . . . . . . . . . . . 16 (π‘Ž = ((β„Žπ‘π‘€)β€˜π‘˜) β†’ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€)) = ((((β„Žπ‘π‘€)β€˜π‘˜)β€˜π‘€)β€˜( 1 β€˜π‘€)))
102 eqid 2725 . . . . . . . . . . . . . . . 16 (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€))) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€)))
103 fvex 6905 . . . . . . . . . . . . . . . 16 ((((β„Žπ‘π‘€)β€˜π‘˜)β€˜π‘€)β€˜( 1 β€˜π‘€)) ∈ V
104101, 102, 103fvmpt 7000 . . . . . . . . . . . . . . 15 (((β„Žπ‘π‘€)β€˜π‘˜) ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) β†’ ((π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€)))β€˜((β„Žπ‘π‘€)β€˜π‘˜)) = ((((β„Žπ‘π‘€)β€˜π‘˜)β€˜π‘€)β€˜( 1 β€˜π‘€)))
10599, 104syl 17 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€)))β€˜((β„Žπ‘π‘€)β€˜π‘˜)) = ((((β„Žπ‘π‘€)β€˜π‘˜)β€˜π‘€)β€˜( 1 β€˜π‘€)))
106 simpr 483 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€))
107 eqid 2725 . . . . . . . . . . . . . . . . 17 (Hom β€˜πΆ) = (Hom β€˜πΆ)
1086, 107, 11, 89, 94catidcl 17661 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ( 1 β€˜π‘€) ∈ (𝑀(Hom β€˜πΆ)𝑀))
10910, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 89, 90, 91, 92, 93, 94, 42, 106, 94, 108yonedalem4b 18267 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((((β„Žπ‘π‘€)β€˜π‘˜)β€˜π‘€)β€˜( 1 β€˜π‘€)) = (((𝑀(2nd β€˜β„Ž)𝑀)β€˜( 1 β€˜π‘€))β€˜π‘˜))
110 eqid 2725 . . . . . . . . . . . . . . . . . 18 (Idβ€˜π‘‚) = (Idβ€˜π‘‚)
111 eqid 2725 . . . . . . . . . . . . . . . . . 18 (Idβ€˜π‘†) = (Idβ€˜π‘†)
112 relfunc 17847 . . . . . . . . . . . . . . . . . . 19 Rel (𝑂 Func 𝑆)
113 1st2ndbr 8044 . . . . . . . . . . . . . . . . . . 19 ((Rel (𝑂 Func 𝑆) ∧ β„Ž ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜β„Ž)(𝑂 Func 𝑆)(2nd β€˜β„Ž))
114112, 93, 113sylancr 585 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ (1st β€˜β„Ž)(𝑂 Func 𝑆)(2nd β€˜β„Ž))
1157, 110, 111, 114, 94funcid 17855 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((𝑀(2nd β€˜β„Ž)𝑀)β€˜((Idβ€˜π‘‚)β€˜π‘€)) = ((Idβ€˜π‘†)β€˜((1st β€˜β„Ž)β€˜π‘€)))
1165, 11oppcid 17702 . . . . . . . . . . . . . . . . . . . 20 (𝐢 ∈ Cat β†’ (Idβ€˜π‘‚) = 1 )
11789, 116syl 17 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ (Idβ€˜π‘‚) = 1 )
118117fveq1d 6894 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((Idβ€˜π‘‚)β€˜π‘€) = ( 1 β€˜π‘€))
119118fveq2d 6896 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((𝑀(2nd β€˜β„Ž)𝑀)β€˜((Idβ€˜π‘‚)β€˜π‘€)) = ((𝑀(2nd β€˜β„Ž)𝑀)β€˜( 1 β€˜π‘€)))
12077ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ π‘ˆ ∈ V)
121 eqid 2725 . . . . . . . . . . . . . . . . . . . . 21 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
1227, 121, 114funcf1 17851 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ (1st β€˜β„Ž):𝐡⟢(Baseβ€˜π‘†))
12312, 120setcbas 18066 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ π‘ˆ = (Baseβ€˜π‘†))
124123feq3d 6704 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((1st β€˜β„Ž):π΅βŸΆπ‘ˆ ↔ (1st β€˜β„Ž):𝐡⟢(Baseβ€˜π‘†)))
125122, 124mpbird 256 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ (1st β€˜β„Ž):π΅βŸΆπ‘ˆ)
126125, 94ffvelcdmd 7090 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((1st β€˜β„Ž)β€˜π‘€) ∈ π‘ˆ)
12712, 111, 120, 126setcid 18074 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((Idβ€˜π‘†)β€˜((1st β€˜β„Ž)β€˜π‘€)) = ( I β†Ύ ((1st β€˜β„Ž)β€˜π‘€)))
128115, 119, 1273eqtr3d 2773 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((𝑀(2nd β€˜β„Ž)𝑀)β€˜( 1 β€˜π‘€)) = ( I β†Ύ ((1st β€˜β„Ž)β€˜π‘€)))
129128fveq1d 6894 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ (((𝑀(2nd β€˜β„Ž)𝑀)β€˜( 1 β€˜π‘€))β€˜π‘˜) = (( I β†Ύ ((1st β€˜β„Ž)β€˜π‘€))β€˜π‘˜))
130 fvresi 7178 . . . . . . . . . . . . . . . 16 (π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€) β†’ (( I β†Ύ ((1st β€˜β„Ž)β€˜π‘€))β€˜π‘˜) = π‘˜)
131130adantl 480 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ (( I β†Ύ ((1st β€˜β„Ž)β€˜π‘€))β€˜π‘˜) = π‘˜)
132109, 129, 1313eqtrd 2769 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((((β„Žπ‘π‘€)β€˜π‘˜)β€˜π‘€)β€˜( 1 β€˜π‘€)) = π‘˜)
13397, 105, 1323eqtrd 2769 . . . . . . . . . . . . 13 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ ((1st β€˜β„Ž)β€˜π‘€)) β†’ ((β„Žπ‘€π‘€)β€˜((β„Žπ‘π‘€)β€˜π‘˜)) = π‘˜)
13488, 133syldan 589 . . . . . . . . . . . 12 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ π‘˜ ∈ (β„Ž(1st β€˜πΈ)𝑀)) β†’ ((β„Žπ‘€π‘€)β€˜((β„Žπ‘π‘€)β€˜π‘˜)) = π‘˜)
135134mpteq2dva 5243 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (π‘˜ ∈ (β„Ž(1st β€˜πΈ)𝑀) ↦ ((β„Žπ‘€π‘€)β€˜((β„Žπ‘π‘€)β€˜π‘˜))) = (π‘˜ ∈ (β„Ž(1st β€˜πΈ)𝑀) ↦ π‘˜))
136 mptresid 6049 . . . . . . . . . . 11 ( I β†Ύ (β„Ž(1st β€˜πΈ)𝑀)) = (π‘˜ ∈ (β„Ž(1st β€˜πΈ)𝑀) ↦ π‘˜)
137135, 136eqtr4di 2783 . . . . . . . . . 10 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (π‘˜ ∈ (β„Ž(1st β€˜πΈ)𝑀) ↦ ((β„Žπ‘€π‘€)β€˜((β„Žπ‘π‘€)β€˜π‘˜))) = ( I β†Ύ (β„Ž(1st β€˜πΈ)𝑀)))
13886, 137eqtrd 2765 . . . . . . . . 9 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘€π‘€) ∘ (β„Žπ‘π‘€)) = ( I β†Ύ (β„Ž(1st β€˜πΈ)𝑀)))
139 fcompt 7138 . . . . . . . . . . 11 (((β„Žπ‘π‘€):(β„Ž(1st β€˜πΈ)𝑀)⟢(β„Ž(1st β€˜π‘)𝑀) ∧ (β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢(β„Ž(1st β€˜πΈ)𝑀)) β†’ ((β„Žπ‘π‘€) ∘ (β„Žπ‘€π‘€)) = (𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀) ↦ ((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))))
14084, 35, 139syl2anc 582 . . . . . . . . . 10 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘π‘€) ∘ (β„Žπ‘€π‘€)) = (𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀) ↦ ((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))))
141 eqid 2725 . . . . . . . . . . . . . 14 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
14228adantr 479 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ 𝐢 ∈ Cat)
14329adantr 479 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ 𝑉 ∈ π‘Š)
14430adantr 479 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
14531adantr 479 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
146 simplrl 775 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ β„Ž ∈ (𝑂 Func 𝑆))
147 simplrr 776 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ 𝑀 ∈ 𝐡)
14881feq3d 6704 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢(β„Ž(1st β€˜πΈ)𝑀) ↔ (β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢((1st β€˜β„Ž)β€˜π‘€)))
14935, 148mpbid 231 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢((1st β€˜β„Ž)β€˜π‘€))
150149ffvelcdmda 7089 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((β„Žπ‘€π‘€)β€˜π‘) ∈ ((1st β€˜β„Ž)β€˜π‘€))
15110, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 142, 143, 144, 145, 146, 147, 42, 150yonedalem4c 18268 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘)) ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž))
152141, 151nat1st2nd 17940 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘)) ∈ (⟨(1st β€˜((1st β€˜π‘Œ)β€˜π‘€)), (2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))⟩(𝑂 Nat 𝑆)⟨(1st β€˜β„Ž), (2nd β€˜β„Ž)⟩))
153141, 152, 7natfn 17943 . . . . . . . . . . . . 13 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘)) Fn 𝐡)
15482eleq2d 2811 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀) ↔ 𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž)))
155154biimpa 475 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ 𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž))
156141, 155nat1st2nd 17940 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ 𝑏 ∈ (⟨(1st β€˜((1st β€˜π‘Œ)β€˜π‘€)), (2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))⟩(𝑂 Nat 𝑆)⟨(1st β€˜β„Ž), (2nd β€˜β„Ž)⟩))
157141, 156, 7natfn 17943 . . . . . . . . . . . . 13 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ 𝑏 Fn 𝐡)
158142adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ 𝐢 ∈ Cat)
159147adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ 𝑀 ∈ 𝐡)
160 simpr 483 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ 𝑧 ∈ 𝐡)
16110, 6, 158, 159, 107, 160yon11 18255 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§) = (𝑧(Hom β€˜πΆ)𝑀))
162161eleq2d 2811 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ (π‘˜ ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§) ↔ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)))
163162biimpa 475 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)) β†’ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀))
164158adantr 479 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ 𝐢 ∈ Cat)
165143ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ 𝑉 ∈ π‘Š)
166144ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
167145ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
168146ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ β„Ž ∈ (𝑂 Func 𝑆))
169159adantr 479 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ 𝑀 ∈ 𝐡)
170150ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((β„Žπ‘€π‘€)β€˜π‘) ∈ ((1st β€˜β„Ž)β€˜π‘€))
171 simplr 767 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ 𝑧 ∈ 𝐡)
172 simpr 483 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀))
17310, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 164, 165, 166, 167, 168, 169, 42, 170, 171, 172yonedalem4b 18267 . . . . . . . . . . . . . . . . 17 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§)β€˜π‘˜) = (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜)β€˜((β„Žπ‘€π‘€)β€˜π‘)))
17410, 6, 11, 5, 12, 13, 3, 14, 1, 15, 16, 164, 165, 166, 167, 168, 169, 26yonedalem3a 18265 . . . . . . . . . . . . . . . . . . . . 21 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((β„Žπ‘€π‘€) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€))) ∧ (β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢(β„Ž(1st β€˜πΈ)𝑀)))
175174simpld 493 . . . . . . . . . . . . . . . . . . . 20 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (β„Žπ‘€π‘€) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€))))
176175fveq1d 6894 . . . . . . . . . . . . . . . . . . 19 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((β„Žπ‘€π‘€)β€˜π‘) = ((π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€)))β€˜π‘))
177155ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ 𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž))
178 fveq1 6891 . . . . . . . . . . . . . . . . . . . . . 22 (π‘Ž = 𝑏 β†’ (π‘Žβ€˜π‘€) = (π‘β€˜π‘€))
179178fveq1d 6894 . . . . . . . . . . . . . . . . . . . . 21 (π‘Ž = 𝑏 β†’ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€)) = ((π‘β€˜π‘€)β€˜( 1 β€˜π‘€)))
180 fvex 6905 . . . . . . . . . . . . . . . . . . . . 21 ((π‘β€˜π‘€)β€˜( 1 β€˜π‘€)) ∈ V
181179, 102, 180fvmpt 7000 . . . . . . . . . . . . . . . . . . . 20 (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) β†’ ((π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€)))β€˜π‘) = ((π‘β€˜π‘€)β€˜( 1 β€˜π‘€)))
182177, 181syl 17 . . . . . . . . . . . . . . . . . . 19 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘€)(𝑂 Nat 𝑆)β„Ž) ↦ ((π‘Žβ€˜π‘€)β€˜( 1 β€˜π‘€)))β€˜π‘) = ((π‘β€˜π‘€)β€˜( 1 β€˜π‘€)))
183176, 182eqtrd 2765 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((β„Žπ‘€π‘€)β€˜π‘) = ((π‘β€˜π‘€)β€˜( 1 β€˜π‘€)))
184183fveq2d 6896 . . . . . . . . . . . . . . . . 17 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜)β€˜((β„Žπ‘€π‘€)β€˜π‘)) = (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜)β€˜((π‘β€˜π‘€)β€˜( 1 β€˜π‘€))))
185156ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ 𝑏 ∈ (⟨(1st β€˜((1st β€˜π‘Œ)β€˜π‘€)), (2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))⟩(𝑂 Nat 𝑆)⟨(1st β€˜β„Ž), (2nd β€˜β„Ž)⟩))
186 eqid 2725 . . . . . . . . . . . . . . . . . . . . . 22 (Hom β€˜π‘‚) = (Hom β€˜π‘‚)
187 eqid 2725 . . . . . . . . . . . . . . . . . . . . . 22 (compβ€˜π‘†) = (compβ€˜π‘†)
188107, 5oppchom 17695 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀(Hom β€˜π‘‚)𝑧) = (𝑧(Hom β€˜πΆ)𝑀)
189172, 188eleqtrrdi 2836 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ π‘˜ ∈ (𝑀(Hom β€˜π‘‚)𝑧))
190141, 185, 7, 186, 187, 169, 171, 189nati 17944 . . . . . . . . . . . . . . . . . . . . 21 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((π‘β€˜π‘§)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€), ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)⟩(compβ€˜π‘†)((1st β€˜β„Ž)β€˜π‘§))((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜)) = (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€), ((1st β€˜β„Ž)β€˜π‘€)⟩(compβ€˜π‘†)((1st β€˜β„Ž)β€˜π‘§))(π‘β€˜π‘€)))
19177ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ π‘ˆ ∈ V)
192191adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ π‘ˆ ∈ V)
193192adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ π‘ˆ ∈ V)
194 relfunc 17847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Rel (𝐢 Func 𝑄)
19510, 17, 5, 12, 3, 77, 19yoncl 18253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (πœ‘ β†’ π‘Œ ∈ (𝐢 Func 𝑄))
196 1st2ndbr 8044 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((Rel (𝐢 Func 𝑄) ∧ π‘Œ ∈ (𝐢 Func 𝑄)) β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
197194, 195, 196sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (πœ‘ β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
1986, 4, 197funcf1 17851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (πœ‘ β†’ (1st β€˜π‘Œ):𝐡⟢(𝑂 Func 𝑆))
199198ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (1st β€˜π‘Œ):𝐡⟢(𝑂 Func 𝑆))
200199, 147ffvelcdmd 7090 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((1st β€˜π‘Œ)β€˜π‘€) ∈ (𝑂 Func 𝑆))
201 1st2ndbr 8044 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Rel (𝑂 Func 𝑆) ∧ ((1st β€˜π‘Œ)β€˜π‘€) ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘€))(𝑂 Func 𝑆)(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€)))
202112, 200, 201sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘€))(𝑂 Func 𝑆)(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€)))
2037, 121, 202funcf1 17851 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘€)):𝐡⟢(Baseβ€˜π‘†))
20412, 191setcbas 18066 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ π‘ˆ = (Baseβ€˜π‘†))
205204feq3d 6704 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€)):π΅βŸΆπ‘ˆ ↔ (1st β€˜((1st β€˜π‘Œ)β€˜π‘€)):𝐡⟢(Baseβ€˜π‘†)))
206203, 205mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘€)):π΅βŸΆπ‘ˆ)
207206, 147ffvelcdmd 7090 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€) ∈ π‘ˆ)
208207ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€) ∈ π‘ˆ)
209206ffvelcdmda 7089 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§) ∈ π‘ˆ)
210209adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§) ∈ π‘ˆ)
211112, 146, 113sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (1st β€˜β„Ž)(𝑂 Func 𝑆)(2nd β€˜β„Ž))
2127, 121, 211funcf1 17851 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (1st β€˜β„Ž):𝐡⟢(Baseβ€˜π‘†))
213204feq3d 6704 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((1st β€˜β„Ž):π΅βŸΆπ‘ˆ ↔ (1st β€˜β„Ž):𝐡⟢(Baseβ€˜π‘†)))
214212, 213mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (1st β€˜β„Ž):π΅βŸΆπ‘ˆ)
215214ffvelcdmda 7089 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ ((1st β€˜β„Ž)β€˜π‘§) ∈ π‘ˆ)
216215adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((1st β€˜β„Ž)β€˜π‘§) ∈ π‘ˆ)
217 eqid 2725 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Hom β€˜π‘†) = (Hom β€˜π‘†)
218202ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘€))(𝑂 Func 𝑆)(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€)))
2197, 186, 217, 218, 169, 171funcf2 17853 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧):(𝑀(Hom β€˜π‘‚)𝑧)⟢(((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)(Hom β€˜π‘†)((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)))
220219, 189ffvelcdmd 7090 . . . . . . . . . . . . . . . . . . . . . . 23 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)(Hom β€˜π‘†)((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)))
22112, 193, 217, 208, 210elsetchom 18069 . . . . . . . . . . . . . . . . . . . . . . 23 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)(Hom β€˜π‘†)((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)) ↔ ((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)))
222220, 221mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§))
223156adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ 𝑏 ∈ (⟨(1st β€˜((1st β€˜π‘Œ)β€˜π‘€)), (2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))⟩(𝑂 Nat 𝑆)⟨(1st β€˜β„Ž), (2nd β€˜β„Ž)⟩))
224141, 223, 7, 217, 160natcl 17942 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ (π‘β€˜π‘§) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)(Hom β€˜π‘†)((1st β€˜β„Ž)β€˜π‘§)))
22512, 192, 217, 209, 215elsetchom 18069 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ ((π‘β€˜π‘§) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)(Hom β€˜π‘†)((1st β€˜β„Ž)β€˜π‘§)) ↔ (π‘β€˜π‘§):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)⟢((1st β€˜β„Ž)β€˜π‘§)))
226224, 225mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ (π‘β€˜π‘§):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)⟢((1st β€˜β„Ž)β€˜π‘§))
227226adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (π‘β€˜π‘§):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)⟢((1st β€˜β„Ž)β€˜π‘§))
22812, 193, 187, 208, 210, 216, 222, 227setcco 18071 . . . . . . . . . . . . . . . . . . . . 21 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((π‘β€˜π‘§)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€), ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)⟩(compβ€˜π‘†)((1st β€˜β„Ž)β€˜π‘§))((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜)) = ((π‘β€˜π‘§) ∘ ((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜)))
229214, 147ffvelcdmd 7090 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((1st β€˜β„Ž)β€˜π‘€) ∈ π‘ˆ)
230229ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((1st β€˜β„Ž)β€˜π‘€) ∈ π‘ˆ)
231141, 156, 7, 217, 147natcl 17942 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (π‘β€˜π‘€) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)(Hom β€˜π‘†)((1st β€˜β„Ž)β€˜π‘€)))
23212, 191, 217, 207, 229elsetchom 18069 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((π‘β€˜π‘€) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)(Hom β€˜π‘†)((1st β€˜β„Ž)β€˜π‘€)) ↔ (π‘β€˜π‘€):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)⟢((1st β€˜β„Ž)β€˜π‘€)))
233231, 232mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ (π‘β€˜π‘€):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)⟢((1st β€˜β„Ž)β€˜π‘€))
234233ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (π‘β€˜π‘€):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€)⟢((1st β€˜β„Ž)β€˜π‘€))
235112, 168, 113sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (1st β€˜β„Ž)(𝑂 Func 𝑆)(2nd β€˜β„Ž))
2367, 186, 217, 235, 169, 171funcf2 17853 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (𝑀(2nd β€˜β„Ž)𝑧):(𝑀(Hom β€˜π‘‚)𝑧)⟢(((1st β€˜β„Ž)β€˜π‘€)(Hom β€˜π‘†)((1st β€˜β„Ž)β€˜π‘§)))
237236, 189ffvelcdmd 7090 . . . . . . . . . . . . . . . . . . . . . . 23 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜) ∈ (((1st β€˜β„Ž)β€˜π‘€)(Hom β€˜π‘†)((1st β€˜β„Ž)β€˜π‘§)))
23812, 193, 217, 230, 216elsetchom 18069 . . . . . . . . . . . . . . . . . . . . . . 23 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜) ∈ (((1st β€˜β„Ž)β€˜π‘€)(Hom β€˜π‘†)((1st β€˜β„Ž)β€˜π‘§)) ↔ ((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜):((1st β€˜β„Ž)β€˜π‘€)⟢((1st β€˜β„Ž)β€˜π‘§)))
239237, 238mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜):((1st β€˜β„Ž)β€˜π‘€)⟢((1st β€˜β„Ž)β€˜π‘§))
24012, 193, 187, 208, 230, 216, 234, 239setcco 18071 . . . . . . . . . . . . . . . . . . . . 21 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€), ((1st β€˜β„Ž)β€˜π‘€)⟩(compβ€˜π‘†)((1st β€˜β„Ž)β€˜π‘§))(π‘β€˜π‘€)) = (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜) ∘ (π‘β€˜π‘€)))
241190, 228, 2403eqtr3d 2773 . . . . . . . . . . . . . . . . . . . 20 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((π‘β€˜π‘§) ∘ ((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜)) = (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜) ∘ (π‘β€˜π‘€)))
242241fveq1d 6894 . . . . . . . . . . . . . . . . . . 19 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (((π‘β€˜π‘§) ∘ ((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜))β€˜( 1 β€˜π‘€)) = ((((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜) ∘ (π‘β€˜π‘€))β€˜( 1 β€˜π‘€)))
2436, 107, 11, 142, 147catidcl 17661 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ( 1 β€˜π‘€) ∈ (𝑀(Hom β€˜πΆ)𝑀))
24410, 6, 142, 147, 107, 147yon11 18255 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€) = (𝑀(Hom β€˜πΆ)𝑀))
245243, 244eleqtrrd 2828 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ( 1 β€˜π‘€) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€))
246245ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ( 1 β€˜π‘€) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘€))
247222, 246fvco3d 6993 . . . . . . . . . . . . . . . . . . 19 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (((π‘β€˜π‘§) ∘ ((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜))β€˜( 1 β€˜π‘€)) = ((π‘β€˜π‘§)β€˜(((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜)β€˜( 1 β€˜π‘€))))
248233, 245fvco3d 6993 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜) ∘ (π‘β€˜π‘€))β€˜( 1 β€˜π‘€)) = (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜)β€˜((π‘β€˜π‘€)β€˜( 1 β€˜π‘€))))
249248ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜) ∘ (π‘β€˜π‘€))β€˜( 1 β€˜π‘€)) = (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜)β€˜((π‘β€˜π‘€)β€˜( 1 β€˜π‘€))))
250242, 247, 2493eqtr3d 2773 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((π‘β€˜π‘§)β€˜(((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜)β€˜( 1 β€˜π‘€))) = (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜)β€˜((π‘β€˜π‘€)β€˜( 1 β€˜π‘€))))
251 eqid 2725 . . . . . . . . . . . . . . . . . . . . 21 (compβ€˜πΆ) = (compβ€˜πΆ)
252243ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ( 1 β€˜π‘€) ∈ (𝑀(Hom β€˜πΆ)𝑀))
25310, 6, 164, 169, 107, 169, 251, 171, 172, 252yon12 18256 . . . . . . . . . . . . . . . . . . . 20 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜)β€˜( 1 β€˜π‘€)) = (( 1 β€˜π‘€)(βŸ¨π‘§, π‘€βŸ©(compβ€˜πΆ)𝑀)π‘˜))
2546, 107, 11, 164, 171, 251, 169, 172catlid 17662 . . . . . . . . . . . . . . . . . . . 20 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (( 1 β€˜π‘€)(βŸ¨π‘§, π‘€βŸ©(compβ€˜πΆ)𝑀)π‘˜) = π‘˜)
255253, 254eqtrd 2765 . . . . . . . . . . . . . . . . . . 19 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜)β€˜( 1 β€˜π‘€)) = π‘˜)
256255fveq2d 6896 . . . . . . . . . . . . . . . . . 18 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((π‘β€˜π‘§)β€˜(((𝑀(2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))𝑧)β€˜π‘˜)β€˜( 1 β€˜π‘€))) = ((π‘β€˜π‘§)β€˜π‘˜))
257250, 256eqtr3d 2767 . . . . . . . . . . . . . . . . 17 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ (((𝑀(2nd β€˜β„Ž)𝑧)β€˜π‘˜)β€˜((π‘β€˜π‘€)β€˜( 1 β€˜π‘€))) = ((π‘β€˜π‘§)β€˜π‘˜))
258173, 184, 2573eqtrd 2769 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ (𝑧(Hom β€˜πΆ)𝑀)) β†’ ((((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§)β€˜π‘˜) = ((π‘β€˜π‘§)β€˜π‘˜))
259163, 258syldan 589 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) ∧ π‘˜ ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)) β†’ ((((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§)β€˜π‘˜) = ((π‘β€˜π‘§)β€˜π‘˜))
260259mpteq2dva 5243 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ (π‘˜ ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§) ↦ ((((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§)β€˜π‘˜)) = (π‘˜ ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§) ↦ ((π‘β€˜π‘§)β€˜π‘˜)))
261152adantr 479 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ ((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘)) ∈ (⟨(1st β€˜((1st β€˜π‘Œ)β€˜π‘€)), (2nd β€˜((1st β€˜π‘Œ)β€˜π‘€))⟩(𝑂 Nat 𝑆)⟨(1st β€˜β„Ž), (2nd β€˜β„Ž)⟩))
262141, 261, 7, 217, 160natcl 17942 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ (((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)(Hom β€˜π‘†)((1st β€˜β„Ž)β€˜π‘§)))
26312, 192, 217, 209, 215elsetchom 18069 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ ((((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)(Hom β€˜π‘†)((1st β€˜β„Ž)β€˜π‘§)) ↔ (((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)⟢((1st β€˜β„Ž)β€˜π‘§)))
264262, 263mpbid 231 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ (((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§):((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§)⟢((1st β€˜β„Ž)β€˜π‘§))
265264feqmptd 6962 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ (((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§) = (π‘˜ ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§) ↦ ((((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§)β€˜π‘˜)))
266226feqmptd 6962 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ (π‘β€˜π‘§) = (π‘˜ ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘€))β€˜π‘§) ↦ ((π‘β€˜π‘§)β€˜π‘˜)))
267260, 265, 2663eqtr4d 2775 . . . . . . . . . . . . 13 ((((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) ∧ 𝑧 ∈ 𝐡) β†’ (((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))β€˜π‘§) = (π‘β€˜π‘§))
268153, 157, 267eqfnfvd 7038 . . . . . . . . . . . 12 (((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) ∧ 𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀)) β†’ ((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘)) = 𝑏)
269268mpteq2dva 5243 . . . . . . . . . . 11 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀) ↦ ((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))) = (𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀) ↦ 𝑏))
270 mptresid 6049 . . . . . . . . . . 11 ( I β†Ύ (β„Ž(1st β€˜π‘)𝑀)) = (𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀) ↦ 𝑏)
271269, 270eqtr4di 2783 . . . . . . . . . 10 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (𝑏 ∈ (β„Ž(1st β€˜π‘)𝑀) ↦ ((β„Žπ‘π‘€)β€˜((β„Žπ‘€π‘€)β€˜π‘))) = ( I β†Ύ (β„Ž(1st β€˜π‘)𝑀)))
272140, 271eqtrd 2765 . . . . . . . . 9 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘π‘€) ∘ (β„Žπ‘€π‘€)) = ( I β†Ύ (β„Ž(1st β€˜π‘)𝑀)))
273 fcof1o 7301 . . . . . . . . 9 ((((β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)⟢(β„Ž(1st β€˜πΈ)𝑀) ∧ (β„Žπ‘π‘€):(β„Ž(1st β€˜πΈ)𝑀)⟢(β„Ž(1st β€˜π‘)𝑀)) ∧ (((β„Žπ‘€π‘€) ∘ (β„Žπ‘π‘€)) = ( I β†Ύ (β„Ž(1st β€˜πΈ)𝑀)) ∧ ((β„Žπ‘π‘€) ∘ (β„Žπ‘€π‘€)) = ( I β†Ύ (β„Ž(1st β€˜π‘)𝑀)))) β†’ ((β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)–1-1-ontoβ†’(β„Ž(1st β€˜πΈ)𝑀) ∧ β—‘(β„Žπ‘€π‘€) = (β„Žπ‘π‘€)))
27435, 84, 138, 272, 273syl22anc 837 . . . . . . . 8 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)–1-1-ontoβ†’(β„Ž(1st β€˜πΈ)𝑀) ∧ β—‘(β„Žπ‘€π‘€) = (β„Žπ‘π‘€)))
275 eqcom 2732 . . . . . . . . 9 (β—‘(β„Žπ‘€π‘€) = (β„Žπ‘π‘€) ↔ (β„Žπ‘π‘€) = β—‘(β„Žπ‘€π‘€))
276275anbi2i 621 . . . . . . . 8 (((β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)–1-1-ontoβ†’(β„Ž(1st β€˜πΈ)𝑀) ∧ β—‘(β„Žπ‘€π‘€) = (β„Žπ‘π‘€)) ↔ ((β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)–1-1-ontoβ†’(β„Ž(1st β€˜πΈ)𝑀) ∧ (β„Žπ‘π‘€) = β—‘(β„Žπ‘€π‘€)))
277274, 276sylib 217 . . . . . . 7 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)–1-1-ontoβ†’(β„Ž(1st β€˜πΈ)𝑀) ∧ (β„Žπ‘π‘€) = β—‘(β„Žπ‘€π‘€)))
278 eqid 2725 . . . . . . . . . . 11 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
279 relfunc 17847 . . . . . . . . . . . 12 Rel ((𝑄 Γ—c 𝑂) Func 𝑇)
280 1st2ndbr 8044 . . . . . . . . . . . 12 ((Rel ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)) β†’ (1st β€˜π‘)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜π‘))
281279, 22, 280sylancr 585 . . . . . . . . . . 11 (πœ‘ β†’ (1st β€˜π‘)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜π‘))
2828, 278, 281funcf1 17851 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜π‘):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
28313, 18setcbas 18066 . . . . . . . . . . 11 (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘‡))
284283feq3d 6704 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜π‘):((𝑂 Func 𝑆) Γ— 𝐡)βŸΆπ‘‰ ↔ (1st β€˜π‘):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡)))
285282, 284mpbird 256 . . . . . . . . 9 (πœ‘ β†’ (1st β€˜π‘):((𝑂 Func 𝑆) Γ— 𝐡)βŸΆπ‘‰)
286285fovcdmda 7589 . . . . . . . 8 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Ž(1st β€˜π‘)𝑀) ∈ 𝑉)
287 1st2ndbr 8044 . . . . . . . . . . . 12 ((Rel ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)) β†’ (1st β€˜πΈ)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜πΈ))
288279, 23, 287sylancr 585 . . . . . . . . . . 11 (πœ‘ β†’ (1st β€˜πΈ)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜πΈ))
2898, 278, 288funcf1 17851 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜πΈ):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
290283feq3d 6704 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜πΈ):((𝑂 Func 𝑆) Γ— 𝐡)βŸΆπ‘‰ ↔ (1st β€˜πΈ):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡)))
291289, 290mpbird 256 . . . . . . . . 9 (πœ‘ β†’ (1st β€˜πΈ):((𝑂 Func 𝑆) Γ— 𝐡)βŸΆπ‘‰)
292291fovcdmda 7589 . . . . . . . 8 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Ž(1st β€˜πΈ)𝑀) ∈ 𝑉)
29313, 29, 286, 292, 25setcinv 18078 . . . . . . 7 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ ((β„Žπ‘€π‘€)((β„Ž(1st β€˜π‘)𝑀)(Invβ€˜π‘‡)(β„Ž(1st β€˜πΈ)𝑀))(β„Žπ‘π‘€) ↔ ((β„Žπ‘€π‘€):(β„Ž(1st β€˜π‘)𝑀)–1-1-ontoβ†’(β„Ž(1st β€˜πΈ)𝑀) ∧ (β„Žπ‘π‘€) = β—‘(β„Žπ‘€π‘€))))
294277, 293mpbird 256 . . . . . 6 ((πœ‘ ∧ (β„Ž ∈ (𝑂 Func 𝑆) ∧ 𝑀 ∈ 𝐡)) β†’ (β„Žπ‘€π‘€)((β„Ž(1st β€˜π‘)𝑀)(Invβ€˜π‘‡)(β„Ž(1st β€˜πΈ)𝑀))(β„Žπ‘π‘€))
295294ralrimivva 3191 . . . . 5 (πœ‘ β†’ βˆ€β„Ž ∈ (𝑂 Func 𝑆)βˆ€π‘€ ∈ 𝐡 (β„Žπ‘€π‘€)((β„Ž(1st β€˜π‘)𝑀)(Invβ€˜π‘‡)(β„Ž(1st β€˜πΈ)𝑀))(β„Žπ‘π‘€))
296 fveq2 6892 . . . . . . . 8 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ (π‘€β€˜π‘§) = (π‘€β€˜βŸ¨β„Ž, π‘€βŸ©))
297 df-ov 7419 . . . . . . . 8 (β„Žπ‘€π‘€) = (π‘€β€˜βŸ¨β„Ž, π‘€βŸ©)
298296, 297eqtr4di 2783 . . . . . . 7 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ (π‘€β€˜π‘§) = (β„Žπ‘€π‘€))
299 fveq2 6892 . . . . . . . . 9 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ ((1st β€˜π‘)β€˜π‘§) = ((1st β€˜π‘)β€˜βŸ¨β„Ž, π‘€βŸ©))
300 df-ov 7419 . . . . . . . . 9 (β„Ž(1st β€˜π‘)𝑀) = ((1st β€˜π‘)β€˜βŸ¨β„Ž, π‘€βŸ©)
301299, 300eqtr4di 2783 . . . . . . . 8 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ ((1st β€˜π‘)β€˜π‘§) = (β„Ž(1st β€˜π‘)𝑀))
302 fveq2 6892 . . . . . . . . 9 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ ((1st β€˜πΈ)β€˜π‘§) = ((1st β€˜πΈ)β€˜βŸ¨β„Ž, π‘€βŸ©))
303 df-ov 7419 . . . . . . . . 9 (β„Ž(1st β€˜πΈ)𝑀) = ((1st β€˜πΈ)β€˜βŸ¨β„Ž, π‘€βŸ©)
304302, 303eqtr4di 2783 . . . . . . . 8 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ ((1st β€˜πΈ)β€˜π‘§) = (β„Ž(1st β€˜πΈ)𝑀))
305301, 304oveq12d 7434 . . . . . . 7 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ (((1st β€˜π‘)β€˜π‘§)(Invβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) = ((β„Ž(1st β€˜π‘)𝑀)(Invβ€˜π‘‡)(β„Ž(1st β€˜πΈ)𝑀)))
306 fveq2 6892 . . . . . . . 8 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ (π‘β€˜π‘§) = (π‘β€˜βŸ¨β„Ž, π‘€βŸ©))
307 df-ov 7419 . . . . . . . 8 (β„Žπ‘π‘€) = (π‘β€˜βŸ¨β„Ž, π‘€βŸ©)
308306, 307eqtr4di 2783 . . . . . . 7 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ (π‘β€˜π‘§) = (β„Žπ‘π‘€))
309298, 305, 308breq123d 5157 . . . . . 6 (𝑧 = βŸ¨β„Ž, π‘€βŸ© β†’ ((π‘€β€˜π‘§)(((1st β€˜π‘)β€˜π‘§)(Invβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘§))(π‘β€˜π‘§) ↔ (β„Žπ‘€π‘€)((β„Ž(1st β€˜π‘)𝑀)(Invβ€˜π‘‡)(β„Ž(1st β€˜πΈ)𝑀))(β„Žπ‘π‘€)))
310309ralxp 5838 . . . . 5 (βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(π‘€β€˜π‘§)(((1st β€˜π‘)β€˜π‘§)(Invβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘§))(π‘β€˜π‘§) ↔ βˆ€β„Ž ∈ (𝑂 Func 𝑆)βˆ€π‘€ ∈ 𝐡 (β„Žπ‘€π‘€)((β„Ž(1st β€˜π‘)𝑀)(Invβ€˜π‘‡)(β„Ž(1st β€˜πΈ)𝑀))(β„Žπ‘π‘€))
311295, 310sylibr 233 . . . 4 (πœ‘ β†’ βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(π‘€β€˜π‘§)(((1st β€˜π‘)β€˜π‘§)(Invβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘§))(π‘β€˜π‘§))
312311r19.21bi 3239 . . 3 ((πœ‘ ∧ 𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡)) β†’ (π‘€β€˜π‘§)(((1st β€˜π‘)β€˜π‘§)(Invβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘§))(π‘β€˜π‘§))
3131, 8, 9, 22, 23, 24, 25, 27, 312invfuc 17965 . 2 (πœ‘ β†’ 𝑀(𝑍𝐼𝐸)(𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ↦ (π‘β€˜π‘§)))
314 fvex 6905 . . . . 5 ((1st β€˜π‘“)β€˜π‘₯) ∈ V
315314mptex 7231 . . . 4 (𝑒 ∈ ((1st β€˜π‘“)β€˜π‘₯) ↦ (𝑦 ∈ 𝐡 ↦ (𝑔 ∈ (𝑦(Hom β€˜πΆ)π‘₯) ↦ (((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘”)β€˜π‘’)))) ∈ V
31642, 315fnmpoi 8072 . . 3 𝑁 Fn ((𝑂 Func 𝑆) Γ— 𝐡)
317 dffn5 6952 . . 3 (𝑁 Fn ((𝑂 Func 𝑆) Γ— 𝐡) ↔ 𝑁 = (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ↦ (π‘β€˜π‘§)))
318316, 317mpbi 229 . 2 𝑁 = (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ↦ (π‘β€˜π‘§))
319313, 318breqtrrdi 5185 1 (πœ‘ β†’ 𝑀(𝑍𝐼𝐸)𝑁)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  Vcvv 3463   βˆͺ cun 3937   βŠ† wss 3939  βŸ¨cop 4630   class class class wbr 5143   ↦ cmpt 5226   I cid 5569   Γ— cxp 5670  β—‘ccnv 5671  ran crn 5673   β†Ύ cres 5674   ∘ ccom 5676  Rel wrel 5677   Fn wfn 6538  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7416   ∈ cmpo 7418  1st c1st 7989  2nd c2nd 7990  tpos ctpos 8229  Basecbs 17179  Hom chom 17243  compcco 17244  Catccat 17643  Idccid 17644  Homf chomf 17645  oppCatcoppc 17690  Invcinv 17727   Func cfunc 17839   ∘func ccofu 17841   Nat cnat 17930   FuncCat cfuc 17931  SetCatcsetc 18063   Γ—c cxpc 18158   1stF c1stf 18159   2ndF c2ndf 18160   ⟨,⟩F cprf 18161   evalF cevlf 18200  HomFchof 18239  Yoncyon 18240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-pm 8846  df-ixp 8915  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-hom 17256  df-cco 17257  df-cat 17647  df-cid 17648  df-homf 17649  df-comf 17650  df-oppc 17691  df-sect 17729  df-inv 17730  df-ssc 17792  df-resc 17793  df-subc 17794  df-func 17843  df-cofu 17845  df-nat 17932  df-fuc 17933  df-setc 18064  df-xpc 18162  df-1stf 18163  df-2ndf 18164  df-prf 18165  df-evlf 18204  df-curf 18205  df-hof 18241  df-yon 18242
This theorem is referenced by:  yonffthlem  18273  yoneda  18274
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