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Theorem yonedalem3 18245
Description: Lemma for yoneda 18248. (Contributed by Mario Carneiro, 28-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‘₯))))
Assertion
Ref Expression
yonedalem3 (πœ‘ β†’ 𝑀 ∈ (𝑍((𝑄 Γ—c 𝑂) Nat 𝑇)𝐸))
Distinct variable groups:   𝑓,π‘Ž,π‘₯, 1   𝐢,π‘Ž,𝑓,π‘₯   𝐸,π‘Ž,𝑓   𝐡,π‘Ž,𝑓,π‘₯   𝑂,π‘Ž,𝑓,π‘₯   𝑆,π‘Ž,𝑓,π‘₯   𝑄,π‘Ž,𝑓,π‘₯   𝑇,𝑓   πœ‘,π‘Ž,𝑓,π‘₯   π‘Œ,π‘Ž,𝑓,π‘₯   𝑍,π‘Ž,𝑓,π‘₯
Allowed substitution hints:   𝑅(π‘₯,𝑓,π‘Ž)   𝑇(π‘₯,π‘Ž)   π‘ˆ(π‘₯,𝑓,π‘Ž)   𝐸(π‘₯)   𝐻(π‘₯,𝑓,π‘Ž)   𝑀(π‘₯,𝑓,π‘Ž)   𝑉(π‘₯,𝑓,π‘Ž)   π‘Š(π‘₯,𝑓,π‘Ž)

Proof of Theorem yonedalem3
Dummy variables 𝑔 𝑦 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.m . . . . 5 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‘₯))))
2 ovex 7438 . . . . . 6 (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ∈ V
32mptex 7220 . . . . 5 (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘₯)(𝑂 Nat 𝑆)𝑓) ↦ ((π‘Žβ€˜π‘₯)β€˜( 1 β€˜π‘₯))) ∈ V
41, 3fnmpoi 8055 . . . 4 𝑀 Fn ((𝑂 Func 𝑆) Γ— 𝐡)
54a1i 11 . . 3 (πœ‘ β†’ 𝑀 Fn ((𝑂 Func 𝑆) Γ— 𝐡))
6 yoneda.y . . . . . . . 8 π‘Œ = (Yonβ€˜πΆ)
7 yoneda.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
8 yoneda.1 . . . . . . . 8 1 = (Idβ€˜πΆ)
9 yoneda.o . . . . . . . 8 𝑂 = (oppCatβ€˜πΆ)
10 yoneda.s . . . . . . . 8 𝑆 = (SetCatβ€˜π‘ˆ)
11 yoneda.t . . . . . . . 8 𝑇 = (SetCatβ€˜π‘‰)
12 yoneda.q . . . . . . . 8 𝑄 = (𝑂 FuncCat 𝑆)
13 yoneda.h . . . . . . . 8 𝐻 = (HomFβ€˜π‘„)
14 yoneda.r . . . . . . . 8 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
15 yoneda.e . . . . . . . 8 𝐸 = (𝑂 evalF 𝑆)
16 yoneda.z . . . . . . . 8 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17 yoneda.c . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ Cat)
1817adantr 480 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝐢 ∈ Cat)
19 yoneda.w . . . . . . . . 9 (πœ‘ β†’ 𝑉 ∈ π‘Š)
2019adantr 480 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝑉 ∈ π‘Š)
21 yoneda.u . . . . . . . . 9 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
2221adantr 480 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
23 yoneda.v . . . . . . . . 9 (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
2423adantr 480 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
25 simprl 768 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝑔 ∈ (𝑂 Func 𝑆))
26 simprr 770 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝑦 ∈ 𝐡)
276, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 1yonedalem3a 18239 . . . . . . 7 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ ((𝑔𝑀𝑦) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘¦)(𝑂 Nat 𝑆)𝑔) ↦ ((π‘Žβ€˜π‘¦)β€˜( 1 β€˜π‘¦))) ∧ (𝑔𝑀𝑦):(𝑔(1st β€˜π‘)𝑦)⟢(𝑔(1st β€˜πΈ)𝑦)))
2827simprd 495 . . . . . 6 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔𝑀𝑦):(𝑔(1st β€˜π‘)𝑦)⟢(𝑔(1st β€˜πΈ)𝑦))
29 eqid 2726 . . . . . . 7 (Hom β€˜π‘‡) = (Hom β€˜π‘‡)
30 eqid 2726 . . . . . . . . . . 11 (𝑄 Γ—c 𝑂) = (𝑄 Γ—c 𝑂)
3112fucbas 17924 . . . . . . . . . . 11 (𝑂 Func 𝑆) = (Baseβ€˜π‘„)
329, 7oppcbas 17672 . . . . . . . . . . 11 𝐡 = (Baseβ€˜π‘‚)
3330, 31, 32xpcbas 18142 . . . . . . . . . 10 ((𝑂 Func 𝑆) Γ— 𝐡) = (Baseβ€˜(𝑄 Γ—c 𝑂))
34 eqid 2726 . . . . . . . . . 10 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
35 relfunc 17821 . . . . . . . . . . 11 Rel ((𝑄 Γ—c 𝑂) Func 𝑇)
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 18237 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)))
3736simpld 494 . . . . . . . . . . 11 (πœ‘ β†’ 𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
38 1st2ndbr 8027 . . . . . . . . . . 11 ((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‘))
3935, 37, 38sylancr 586 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜π‘)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜π‘))
4033, 34, 39funcf1 17825 . . . . . . . . 9 (πœ‘ β†’ (1st β€˜π‘):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
4140fovcdmda 7575 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔(1st β€˜π‘)𝑦) ∈ (Baseβ€˜π‘‡))
4211, 20setcbas 18040 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ 𝑉 = (Baseβ€˜π‘‡))
4341, 42eleqtrrd 2830 . . . . . . 7 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔(1st β€˜π‘)𝑦) ∈ 𝑉)
4436simprd 495 . . . . . . . . . . 11 (πœ‘ β†’ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
45 1st2ndbr 8027 . . . . . . . . . . 11 ((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))
4635, 44, 45sylancr 586 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜πΈ)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜πΈ))
4733, 34, 46funcf1 17825 . . . . . . . . 9 (πœ‘ β†’ (1st β€˜πΈ):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
4847fovcdmda 7575 . . . . . . . 8 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔(1st β€˜πΈ)𝑦) ∈ (Baseβ€˜π‘‡))
4948, 42eleqtrrd 2830 . . . . . . 7 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔(1st β€˜πΈ)𝑦) ∈ 𝑉)
5011, 20, 29, 43, 49elsetchom 18043 . . . . . 6 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ ((𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)) ↔ (𝑔𝑀𝑦):(𝑔(1st β€˜π‘)𝑦)⟢(𝑔(1st β€˜πΈ)𝑦)))
5128, 50mpbird 257 . . . . 5 ((πœ‘ ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐡)) β†’ (𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)))
5251ralrimivva 3194 . . . 4 (πœ‘ β†’ βˆ€π‘” ∈ (𝑂 Func 𝑆)βˆ€π‘¦ ∈ 𝐡 (𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)))
53 fveq2 6885 . . . . . . 7 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ (π‘€β€˜π‘§) = (π‘€β€˜βŸ¨π‘”, π‘¦βŸ©))
54 df-ov 7408 . . . . . . 7 (𝑔𝑀𝑦) = (π‘€β€˜βŸ¨π‘”, π‘¦βŸ©)
5553, 54eqtr4di 2784 . . . . . 6 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ (π‘€β€˜π‘§) = (𝑔𝑀𝑦))
56 fveq2 6885 . . . . . . . 8 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((1st β€˜π‘)β€˜π‘§) = ((1st β€˜π‘)β€˜βŸ¨π‘”, π‘¦βŸ©))
57 df-ov 7408 . . . . . . . 8 (𝑔(1st β€˜π‘)𝑦) = ((1st β€˜π‘)β€˜βŸ¨π‘”, π‘¦βŸ©)
5856, 57eqtr4di 2784 . . . . . . 7 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((1st β€˜π‘)β€˜π‘§) = (𝑔(1st β€˜π‘)𝑦))
59 fveq2 6885 . . . . . . . 8 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((1st β€˜πΈ)β€˜π‘§) = ((1st β€˜πΈ)β€˜βŸ¨π‘”, π‘¦βŸ©))
60 df-ov 7408 . . . . . . . 8 (𝑔(1st β€˜πΈ)𝑦) = ((1st β€˜πΈ)β€˜βŸ¨π‘”, π‘¦βŸ©)
6159, 60eqtr4di 2784 . . . . . . 7 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((1st β€˜πΈ)β€˜π‘§) = (𝑔(1st β€˜πΈ)𝑦))
6258, 61oveq12d 7423 . . . . . 6 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) = ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)))
6355, 62eleq12d 2821 . . . . 5 (𝑧 = βŸ¨π‘”, π‘¦βŸ© β†’ ((π‘€β€˜π‘§) ∈ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) ↔ (𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦))))
6463ralxp 5835 . . . 4 (βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(π‘€β€˜π‘§) ∈ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) ↔ βˆ€π‘” ∈ (𝑂 Func 𝑆)βˆ€π‘¦ ∈ 𝐡 (𝑔𝑀𝑦) ∈ ((𝑔(1st β€˜π‘)𝑦)(Hom β€˜π‘‡)(𝑔(1st β€˜πΈ)𝑦)))
6552, 64sylibr 233 . . 3 (πœ‘ β†’ βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(π‘€β€˜π‘§) ∈ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)))
66 ovex 7438 . . . . . 6 (𝑂 Func 𝑆) ∈ V
677fvexi 6899 . . . . . 6 𝐡 ∈ V
6866, 67mpoex 8065 . . . . 5 (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‘₯)))) ∈ V
691, 68eqeltri 2823 . . . 4 𝑀 ∈ V
7069elixp 8900 . . 3 (𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) ↔ (𝑀 Fn ((𝑂 Func 𝑆) Γ— 𝐡) ∧ βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(π‘€β€˜π‘§) ∈ (((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§))))
715, 65, 70sylanbrc 582 . 2 (πœ‘ β†’ 𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)))
7217adantr 480 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝐢 ∈ Cat)
7319adantr 480 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑉 ∈ π‘Š)
7421adantr 480 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
7523adantr 480 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
76 simpr1 1191 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
77 xp1st 8006 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ (1st β€˜π‘§) ∈ (𝑂 Func 𝑆))
7876, 77syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (1st β€˜π‘§) ∈ (𝑂 Func 𝑆))
79 xp2nd 8007 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ (2nd β€˜π‘§) ∈ 𝐡)
8076, 79syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (2nd β€˜π‘§) ∈ 𝐡)
81 simpr2 1192 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
82 xp1st 8006 . . . . . 6 (𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ (1st β€˜π‘€) ∈ (𝑂 Func 𝑆))
8381, 82syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (1st β€˜π‘€) ∈ (𝑂 Func 𝑆))
84 xp2nd 8007 . . . . . 6 (𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ (2nd β€˜π‘€) ∈ 𝐡)
8581, 84syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (2nd β€˜π‘€) ∈ 𝐡)
86 simpr3 1193 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))
87 eqid 2726 . . . . . . . . . 10 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8812, 87fuchom 17925 . . . . . . . . 9 (𝑂 Nat 𝑆) = (Hom β€˜π‘„)
89 eqid 2726 . . . . . . . . 9 (Hom β€˜π‘‚) = (Hom β€˜π‘‚)
90 eqid 2726 . . . . . . . . 9 (Hom β€˜(𝑄 Γ—c 𝑂)) = (Hom β€˜(𝑄 Γ—c 𝑂))
9130, 33, 88, 89, 90, 76, 81xpchom 18144 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀) = (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘§)(Hom β€˜π‘‚)(2nd β€˜π‘€))))
92 eqid 2726 . . . . . . . . . 10 (Hom β€˜πΆ) = (Hom β€˜πΆ)
9392, 9oppchom 17669 . . . . . . . . 9 ((2nd β€˜π‘§)(Hom β€˜π‘‚)(2nd β€˜π‘€)) = ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))
9493xpeq2i 5696 . . . . . . . 8 (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘§)(Hom β€˜π‘‚)(2nd β€˜π‘€))) = (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§)))
9591, 94eqtrdi 2782 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀) = (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))))
9686, 95eleqtrd 2829 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑔 ∈ (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))))
97 xp1st 8006 . . . . . 6 (𝑔 ∈ (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))) β†’ (1st β€˜π‘”) ∈ ((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)))
9896, 97syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (1st β€˜π‘”) ∈ ((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)))
99 xp2nd 8007 . . . . . 6 (𝑔 ∈ (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))) β†’ (2nd β€˜π‘”) ∈ ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§)))
10096, 99syl 17 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (2nd β€˜π‘”) ∈ ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§)))
1016, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 72, 73, 74, 75, 78, 80, 83, 85, 98, 100, 1yonedalem3b 18244 . . . 4 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (((1st β€˜π‘€)𝑀(2nd β€˜π‘€))(⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”))) = (((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”))(⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))((1st β€˜π‘§)𝑀(2nd β€˜π‘§))))
102 1st2nd2 8013 . . . . . . . . . 10 (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
10376, 102syl 17 . . . . . . . . 9 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
104103fveq2d 6889 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜π‘)β€˜π‘§) = ((1st β€˜π‘)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
105 df-ov 7408 . . . . . . . 8 ((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)) = ((1st β€˜π‘)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
106104, 105eqtr4di 2784 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜π‘)β€˜π‘§) = ((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)))
107 1st2nd2 8013 . . . . . . . . . 10 (𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
10881, 107syl 17 . . . . . . . . 9 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
109108fveq2d 6889 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜π‘)β€˜π‘€) = ((1st β€˜π‘)β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
110 df-ov 7408 . . . . . . . 8 ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€)) = ((1st β€˜π‘)β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
111109, 110eqtr4di 2784 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜π‘)β€˜π‘€) = ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€)))
112106, 111opeq12d 4876 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩ = ⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€))⟩)
113108fveq2d 6889 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜πΈ)β€˜π‘€) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
114 df-ov 7408 . . . . . . 7 ((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
115113, 114eqtr4di 2784 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜πΈ)β€˜π‘€) = ((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))
116112, 115oveq12d 7423 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€)) = (⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€))))
117108fveq2d 6889 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (π‘€β€˜π‘€) = (π‘€β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
118 df-ov 7408 . . . . . 6 ((1st β€˜π‘€)𝑀(2nd β€˜π‘€)) = (π‘€β€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
119117, 118eqtr4di 2784 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (π‘€β€˜π‘€) = ((1st β€˜π‘€)𝑀(2nd β€˜π‘€)))
120103, 108oveq12d 7423 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (𝑧(2nd β€˜π‘)𝑀) = (⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
121 1st2nd2 8013 . . . . . . . 8 (𝑔 ∈ (((1st β€˜π‘§)(𝑂 Nat 𝑆)(1st β€˜π‘€)) Γ— ((2nd β€˜π‘€)(Hom β€˜πΆ)(2nd β€˜π‘§))) β†’ 𝑔 = ⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)
12296, 121syl 17 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ 𝑔 = ⟨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)
123120, 122fveq12d 6892 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”) = ((⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)β€˜βŸ¨(1st β€˜π‘”), (2nd β€˜π‘”)⟩))
124 df-ov 7408 . . . . . 6 ((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”)) = ((⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)β€˜βŸ¨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)
125123, 124eqtr4di 2784 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”) = ((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”)))
126116, 119, 125oveq123d 7426 . . . 4 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((π‘€β€˜π‘€)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”)) = (((1st β€˜π‘€)𝑀(2nd β€˜π‘€))(⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘€)(1st β€˜π‘)(2nd β€˜π‘€))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜π‘)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”))))
127103fveq2d 6889 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜πΈ)β€˜π‘§) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
128 df-ov 7408 . . . . . . . 8 ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§)) = ((1st β€˜πΈ)β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
129127, 128eqtr4di 2784 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((1st β€˜πΈ)β€˜π‘§) = ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§)))
130106, 129opeq12d 4876 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩ = ⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§))⟩)
131130, 115oveq12d 7423 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€)) = (⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€))))
132103, 108oveq12d 7423 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (𝑧(2nd β€˜πΈ)𝑀) = (⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
133132, 122fveq12d 6892 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”) = ((⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)β€˜βŸ¨(1st β€˜π‘”), (2nd β€˜π‘”)⟩))
134 df-ov 7408 . . . . . 6 ((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”)) = ((⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)β€˜βŸ¨(1st β€˜π‘”), (2nd β€˜π‘”)⟩)
135133, 134eqtr4di 2784 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”) = ((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”)))
136103fveq2d 6889 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (π‘€β€˜π‘§) = (π‘€β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
137 df-ov 7408 . . . . . 6 ((1st β€˜π‘§)𝑀(2nd β€˜π‘§)) = (π‘€β€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
138136, 137eqtr4di 2784 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (π‘€β€˜π‘§) = ((1st β€˜π‘§)𝑀(2nd β€˜π‘§)))
139131, 135, 138oveq123d 7426 . . . 4 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ (((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))(π‘€β€˜π‘§)) = (((1st β€˜π‘”)(⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩(2nd β€˜πΈ)⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)(2nd β€˜π‘”))(⟨((1st β€˜π‘§)(1st β€˜π‘)(2nd β€˜π‘§)), ((1st β€˜π‘§)(1st β€˜πΈ)(2nd β€˜π‘§))⟩(compβ€˜π‘‡)((1st β€˜π‘€)(1st β€˜πΈ)(2nd β€˜π‘€)))((1st β€˜π‘§)𝑀(2nd β€˜π‘§))))
140101, 126, 1393eqtr4d 2776 . . 3 ((πœ‘ ∧ (𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑀 ∈ ((𝑂 Func 𝑆) Γ— 𝐡) ∧ 𝑔 ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀))) β†’ ((π‘€β€˜π‘€)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”)) = (((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))(π‘€β€˜π‘§)))
141140ralrimivvva 3197 . 2 (πœ‘ β†’ βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)βˆ€π‘€ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)βˆ€π‘” ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀)((π‘€β€˜π‘€)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”)) = (((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))(π‘€β€˜π‘§)))
142 eqid 2726 . . 3 ((𝑄 Γ—c 𝑂) Nat 𝑇) = ((𝑄 Γ—c 𝑂) Nat 𝑇)
143 eqid 2726 . . 3 (compβ€˜π‘‡) = (compβ€˜π‘‡)
144142, 33, 90, 29, 143, 37, 44isnat2 17911 . 2 (πœ‘ β†’ (𝑀 ∈ (𝑍((𝑄 Γ—c 𝑂) Nat 𝑇)𝐸) ↔ (𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) Γ— 𝐡)(((1st β€˜π‘)β€˜π‘§)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜π‘§)) ∧ βˆ€π‘§ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)βˆ€π‘€ ∈ ((𝑂 Func 𝑆) Γ— 𝐡)βˆ€π‘” ∈ (𝑧(Hom β€˜(𝑄 Γ—c 𝑂))𝑀)((π‘€β€˜π‘€)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜π‘)β€˜π‘€)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))((𝑧(2nd β€˜π‘)𝑀)β€˜π‘”)) = (((𝑧(2nd β€˜πΈ)𝑀)β€˜π‘”)(⟨((1st β€˜π‘)β€˜π‘§), ((1st β€˜πΈ)β€˜π‘§)⟩(compβ€˜π‘‡)((1st β€˜πΈ)β€˜π‘€))(π‘€β€˜π‘§)))))
14571, 141, 144mpbir2and 710 1 (πœ‘ β†’ 𝑀 ∈ (𝑍((𝑄 Γ—c 𝑂) Nat 𝑇)𝐸))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   βˆͺ cun 3941   βŠ† wss 3943  βŸ¨cop 4629   class class class wbr 5141   ↦ cmpt 5224   Γ— cxp 5667  ran crn 5670  Rel wrel 5674   Fn wfn 6532  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7972  2nd c2nd 7973  tpos ctpos 8211  Xcixp 8893  Basecbs 17153  Hom chom 17217  compcco 17218  Catccat 17617  Idccid 17618  Homf chomf 17619  oppCatcoppc 17664   Func cfunc 17813   ∘func ccofu 17815   Nat cnat 17904   FuncCat cfuc 17905  SetCatcsetc 18037   Γ—c cxpc 18132   1stF c1stf 18133   2ndF c2ndf 18134   ⟨,⟩F cprf 18135   evalF cevlf 18174  HomFchof 18213  Yoncyon 18214
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-hom 17230  df-cco 17231  df-cat 17621  df-cid 17622  df-homf 17623  df-comf 17624  df-oppc 17665  df-ssc 17766  df-resc 17767  df-subc 17768  df-func 17817  df-cofu 17819  df-nat 17906  df-fuc 17907  df-setc 18038  df-xpc 18136  df-1stf 18137  df-2ndf 18138  df-prf 18139  df-evlf 18178  df-curf 18179  df-hof 18215  df-yon 18216
This theorem is referenced by:  yonedainv  18246
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