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Theorem yonedalem3b 18265
Description: Lemma for yoneda 18269. (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)
yonedalem21.f (πœ‘ β†’ 𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
yonedalem22.g (πœ‘ β†’ 𝐺 ∈ (𝑂 Func 𝑆))
yonedalem22.p (πœ‘ β†’ 𝑃 ∈ 𝐡)
yonedalem22.a (πœ‘ β†’ 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
yonedalem22.k (πœ‘ β†’ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋))
yonedalem3.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
yonedalem3b (πœ‘ β†’ ((𝐺𝑀𝑃)(⟨(𝐹(1st β€˜π‘)𝑋), (𝐺(1st β€˜π‘)𝑃)⟩(compβ€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃))(𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾)) = ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾)(⟨(𝐹(1st β€˜π‘)𝑋), (𝐹(1st β€˜πΈ)𝑋)⟩(compβ€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃))(𝐹𝑀𝑋)))
Distinct variable groups:   𝑓,π‘Ž,π‘₯, 1   𝐴,π‘Ž   𝐢,π‘Ž,𝑓,π‘₯   𝐸,π‘Ž,𝑓   𝐹,π‘Ž,𝑓,π‘₯   𝐾,π‘Ž   𝐡,π‘Ž,𝑓,π‘₯   𝐺,π‘Ž,𝑓,π‘₯   𝑂,π‘Ž,𝑓,π‘₯   𝑆,π‘Ž,𝑓,π‘₯   𝑄,π‘Ž,𝑓,π‘₯   𝑇,𝑓   𝑃,π‘Ž,𝑓,π‘₯   πœ‘,π‘Ž,𝑓,π‘₯   π‘Œ,π‘Ž,𝑓,π‘₯   𝑍,π‘Ž,𝑓,π‘₯   𝑋,π‘Ž,𝑓,π‘₯
Allowed substitution hints:   𝐴(π‘₯,𝑓)   𝑅(π‘₯,𝑓,π‘Ž)   𝑇(π‘₯,π‘Ž)   π‘ˆ(π‘₯,𝑓,π‘Ž)   𝐸(π‘₯)   𝐻(π‘₯,𝑓,π‘Ž)   𝐾(π‘₯,𝑓)   𝑀(π‘₯,𝑓,π‘Ž)   𝑉(π‘₯,𝑓,π‘Ž)   π‘Š(π‘₯,𝑓,π‘Ž)

Proof of Theorem yonedalem3b
Dummy variables 𝑏 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7421 . . . . . . . 8 (𝑏 = π‘Ž β†’ (𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏) = (𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž))
21oveq1d 7428 . . . . . . 7 (𝑏 = π‘Ž β†’ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)) = ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)))
32fveq1d 6892 . . . . . 6 (𝑏 = π‘Ž β†’ (((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ) = (((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ))
43fveq1d 6892 . . . . 5 (𝑏 = π‘Ž β†’ ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)) = ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)))
54cbvmptv 5257 . . . 4 (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)))
6 yoneda.q . . . . . . . . 9 𝑄 = (𝑂 FuncCat 𝑆)
7 eqid 2725 . . . . . . . . 9 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8 yoneda.o . . . . . . . . . 10 𝑂 = (oppCatβ€˜πΆ)
9 yoneda.b . . . . . . . . . 10 𝐡 = (Baseβ€˜πΆ)
108, 9oppcbas 17693 . . . . . . . . 9 𝐡 = (Baseβ€˜π‘‚)
11 eqid 2725 . . . . . . . . 9 (compβ€˜π‘†) = (compβ€˜π‘†)
12 eqid 2725 . . . . . . . . 9 (compβ€˜π‘„) = (compβ€˜π‘„)
13 eqid 2725 . . . . . . . . . . . 12 (Hom β€˜πΆ) = (Hom β€˜πΆ)
146, 7fuchom 17946 . . . . . . . . . . . 12 (𝑂 Nat 𝑆) = (Hom β€˜π‘„)
15 relfunc 17842 . . . . . . . . . . . . 13 Rel (𝐢 Func 𝑄)
16 yoneda.y . . . . . . . . . . . . . 14 π‘Œ = (Yonβ€˜πΆ)
17 yoneda.c . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐢 ∈ Cat)
18 yoneda.s . . . . . . . . . . . . . 14 𝑆 = (SetCatβ€˜π‘ˆ)
19 yoneda.w . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑉 ∈ π‘Š)
20 yoneda.v . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (ran (Homf β€˜π‘„) βˆͺ π‘ˆ) βŠ† 𝑉)
2120unssbd 4183 . . . . . . . . . . . . . . 15 (πœ‘ β†’ π‘ˆ βŠ† 𝑉)
2219, 21ssexd 5320 . . . . . . . . . . . . . 14 (πœ‘ β†’ π‘ˆ ∈ V)
23 yoneda.u . . . . . . . . . . . . . 14 (πœ‘ β†’ ran (Homf β€˜πΆ) βŠ† π‘ˆ)
2416, 17, 8, 18, 6, 22, 23yoncl 18248 . . . . . . . . . . . . 13 (πœ‘ β†’ π‘Œ ∈ (𝐢 Func 𝑄))
25 1st2ndbr 8040 . . . . . . . . . . . . 13 ((Rel (𝐢 Func 𝑄) ∧ π‘Œ ∈ (𝐢 Func 𝑄)) β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
2615, 24, 25sylancr 585 . . . . . . . . . . . 12 (πœ‘ β†’ (1st β€˜π‘Œ)(𝐢 Func 𝑄)(2nd β€˜π‘Œ))
27 yonedalem22.p . . . . . . . . . . . 12 (πœ‘ β†’ 𝑃 ∈ 𝐡)
28 yonedalem21.x . . . . . . . . . . . 12 (πœ‘ β†’ 𝑋 ∈ 𝐡)
299, 13, 14, 26, 27, 28funcf2 17848 . . . . . . . . . . 11 (πœ‘ β†’ (𝑃(2nd β€˜π‘Œ)𝑋):(𝑃(Hom β€˜πΆ)𝑋)⟢(((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)((1st β€˜π‘Œ)β€˜π‘‹)))
30 yonedalem22.k . . . . . . . . . . 11 (πœ‘ β†’ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋))
3129, 30ffvelcdmd 7088 . . . . . . . . . 10 (πœ‘ β†’ ((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ) ∈ (((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)((1st β€˜π‘Œ)β€˜π‘‹)))
3231adantr 479 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ) ∈ (((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)((1st β€˜π‘Œ)β€˜π‘‹)))
33 simpr 483 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹))
34 yonedalem22.a . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
3534adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
366, 7, 12, 33, 35fuccocl 17950 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž) ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐺))
3727adantr 479 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝑃 ∈ 𝐡)
386, 7, 10, 11, 12, 32, 36, 37fuccoval 17949 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ) = (((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)β€˜π‘ƒ)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ), ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ))(((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)))
396, 7, 10, 11, 12, 33, 35, 37fuccoval 17949 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)β€˜π‘ƒ) = ((π΄β€˜π‘ƒ)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ), ((1st β€˜πΉ)β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ))(π‘Žβ€˜π‘ƒ)))
4022adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ π‘ˆ ∈ V)
41 eqid 2725 . . . . . . . . . . . . . . 15 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
42 relfunc 17842 . . . . . . . . . . . . . . . 16 Rel (𝑂 Func 𝑆)
436fucbas 17945 . . . . . . . . . . . . . . . . . 18 (𝑂 Func 𝑆) = (Baseβ€˜π‘„)
449, 43, 26funcf1 17846 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (1st β€˜π‘Œ):𝐡⟢(𝑂 Func 𝑆))
4544, 28ffvelcdmd 7088 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ ((1st β€˜π‘Œ)β€˜π‘‹) ∈ (𝑂 Func 𝑆))
46 1st2ndbr 8040 . . . . . . . . . . . . . . . 16 ((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‘‹)))
4742, 45, 46sylancr 585 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))(𝑂 Func 𝑆)(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹)))
4810, 41, 47funcf1 17846 . . . . . . . . . . . . . 14 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)):𝐡⟢(Baseβ€˜π‘†))
4918, 22setcbas 18061 . . . . . . . . . . . . . . 15 (πœ‘ β†’ π‘ˆ = (Baseβ€˜π‘†))
5049feq3d 6704 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)):π΅βŸΆπ‘ˆ ↔ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)):𝐡⟢(Baseβ€˜π‘†)))
5148, 50mpbird 256 . . . . . . . . . . . . 13 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)):π΅βŸΆπ‘ˆ)
5251, 27ffvelcdmd 7088 . . . . . . . . . . . 12 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ) ∈ π‘ˆ)
5352adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ) ∈ π‘ˆ)
54 yonedalem21.f . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝐹 ∈ (𝑂 Func 𝑆))
55 1st2ndbr 8040 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜πΉ)(𝑂 Func 𝑆)(2nd β€˜πΉ))
5642, 54, 55sylancr 585 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (1st β€˜πΉ)(𝑂 Func 𝑆)(2nd β€˜πΉ))
5710, 41, 56funcf1 17846 . . . . . . . . . . . . . 14 (πœ‘ β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π‘†))
5849feq3d 6704 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((1st β€˜πΉ):π΅βŸΆπ‘ˆ ↔ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π‘†)))
5957, 58mpbird 256 . . . . . . . . . . . . 13 (πœ‘ β†’ (1st β€˜πΉ):π΅βŸΆπ‘ˆ)
6059, 27ffvelcdmd 7088 . . . . . . . . . . . 12 (πœ‘ β†’ ((1st β€˜πΉ)β€˜π‘ƒ) ∈ π‘ˆ)
6160adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((1st β€˜πΉ)β€˜π‘ƒ) ∈ π‘ˆ)
62 yonedalem22.g . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝐺 ∈ (𝑂 Func 𝑆))
63 1st2ndbr 8040 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐺 ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜πΊ)(𝑂 Func 𝑆)(2nd β€˜πΊ))
6442, 62, 63sylancr 585 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (1st β€˜πΊ)(𝑂 Func 𝑆)(2nd β€˜πΊ))
6510, 41, 64funcf1 17846 . . . . . . . . . . . . . 14 (πœ‘ β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π‘†))
6665, 27ffvelcdmd 7088 . . . . . . . . . . . . 13 (πœ‘ β†’ ((1st β€˜πΊ)β€˜π‘ƒ) ∈ (Baseβ€˜π‘†))
6766, 49eleqtrrd 2828 . . . . . . . . . . . 12 (πœ‘ β†’ ((1st β€˜πΊ)β€˜π‘ƒ) ∈ π‘ˆ)
6867adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((1st β€˜πΊ)β€˜π‘ƒ) ∈ π‘ˆ)
697, 33nat1st2nd 17935 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ π‘Ž ∈ (⟨(1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)), (2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))⟩(𝑂 Nat 𝑆)⟨(1st β€˜πΉ), (2nd β€˜πΉ)⟩))
70 eqid 2725 . . . . . . . . . . . . 13 (Hom β€˜π‘†) = (Hom β€˜π‘†)
717, 69, 10, 70, 37natcl 17937 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (π‘Žβ€˜π‘ƒ) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)(Hom β€˜π‘†)((1st β€˜πΉ)β€˜π‘ƒ)))
7218, 40, 70, 53, 61elsetchom 18064 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘ƒ) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)(Hom β€˜π‘†)((1st β€˜πΉ)β€˜π‘ƒ)) ↔ (π‘Žβ€˜π‘ƒ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟢((1st β€˜πΉ)β€˜π‘ƒ)))
7371, 72mpbid 231 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (π‘Žβ€˜π‘ƒ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟢((1st β€˜πΉ)β€˜π‘ƒ))
747, 34nat1st2nd 17935 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐴 ∈ (⟨(1st β€˜πΉ), (2nd β€˜πΉ)⟩(𝑂 Nat 𝑆)⟨(1st β€˜πΊ), (2nd β€˜πΊ)⟩))
757, 74, 10, 70, 27natcl 17937 . . . . . . . . . . . . 13 (πœ‘ β†’ (π΄β€˜π‘ƒ) ∈ (((1st β€˜πΉ)β€˜π‘ƒ)(Hom β€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ)))
7618, 22, 70, 60, 67elsetchom 18064 . . . . . . . . . . . . 13 (πœ‘ β†’ ((π΄β€˜π‘ƒ) ∈ (((1st β€˜πΉ)β€˜π‘ƒ)(Hom β€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ)) ↔ (π΄β€˜π‘ƒ):((1st β€˜πΉ)β€˜π‘ƒ)⟢((1st β€˜πΊ)β€˜π‘ƒ)))
7775, 76mpbid 231 . . . . . . . . . . . 12 (πœ‘ β†’ (π΄β€˜π‘ƒ):((1st β€˜πΉ)β€˜π‘ƒ)⟢((1st β€˜πΊ)β€˜π‘ƒ))
7877adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (π΄β€˜π‘ƒ):((1st β€˜πΉ)β€˜π‘ƒ)⟢((1st β€˜πΊ)β€˜π‘ƒ))
7918, 40, 11, 53, 61, 68, 73, 78setcco 18066 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π΄β€˜π‘ƒ)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ), ((1st β€˜πΉ)β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ))(π‘Žβ€˜π‘ƒ)) = ((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ)))
8039, 79eqtrd 2765 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)β€˜π‘ƒ) = ((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ)))
8180oveq1d 7428 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)β€˜π‘ƒ)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ), ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ))(((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)) = (((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ))(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ), ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ))(((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)))
8244, 27ffvelcdmd 7088 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((1st β€˜π‘Œ)β€˜π‘ƒ) ∈ (𝑂 Func 𝑆))
83 1st2ndbr 8040 . . . . . . . . . . . . . 14 ((Rel (𝑂 Func 𝑆) ∧ ((1st β€˜π‘Œ)β€˜π‘ƒ) ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))(𝑂 Func 𝑆)(2nd β€˜((1st β€˜π‘Œ)β€˜π‘ƒ)))
8442, 82, 83sylancr 585 . . . . . . . . . . . . 13 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))(𝑂 Func 𝑆)(2nd β€˜((1st β€˜π‘Œ)β€˜π‘ƒ)))
8510, 41, 84funcf1 17846 . . . . . . . . . . . 12 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ)):𝐡⟢(Baseβ€˜π‘†))
8685, 27ffvelcdmd 7088 . . . . . . . . . . 11 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ) ∈ (Baseβ€˜π‘†))
8786, 49eleqtrrd 2828 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ) ∈ π‘ˆ)
8887adantr 479 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ) ∈ π‘ˆ)
897, 31nat1st2nd 17935 . . . . . . . . . . . 12 (πœ‘ β†’ ((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ) ∈ (⟨(1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ)), (2nd β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))⟩(𝑂 Nat 𝑆)⟨(1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)), (2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))⟩))
907, 89, 10, 70, 27natcl 17937 . . . . . . . . . . 11 (πœ‘ β†’ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ)(Hom β€˜π‘†)((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)))
9118, 22, 70, 87, 52elsetchom 18064 . . . . . . . . . . 11 (πœ‘ β†’ ((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ)(Hom β€˜π‘†)((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)) ↔ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)))
9290, 91mpbid 231 . . . . . . . . . 10 (πœ‘ β†’ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ))
9392adantr 479 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ))
94 fco 6741 . . . . . . . . . 10 (((π΄β€˜π‘ƒ):((1st β€˜πΉ)β€˜π‘ƒ)⟢((1st β€˜πΊ)β€˜π‘ƒ) ∧ (π‘Žβ€˜π‘ƒ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟢((1st β€˜πΉ)β€˜π‘ƒ)) β†’ ((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ)):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟢((1st β€˜πΊ)β€˜π‘ƒ))
9578, 73, 94syl2anc 582 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ)):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟢((1st β€˜πΊ)β€˜π‘ƒ))
9618, 40, 11, 88, 53, 68, 93, 95setcco 18066 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ))(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ), ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ))(((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)) = (((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ)) ∘ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)))
9738, 81, 963eqtrd 2769 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ) = (((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ)) ∘ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)))
9897fveq1d 6892 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)) = ((((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ)) ∘ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ))β€˜( 1 β€˜π‘ƒ)))
99 yoneda.1 . . . . . . . . . 10 1 = (Idβ€˜πΆ)
1009, 13, 99, 17, 27catidcl 17656 . . . . . . . . 9 (πœ‘ β†’ ( 1 β€˜π‘ƒ) ∈ (𝑃(Hom β€˜πΆ)𝑃))
10116, 9, 17, 27, 13, 27yon11 18250 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ) = (𝑃(Hom β€˜πΆ)𝑃))
102100, 101eleqtrrd 2828 . . . . . . . 8 (πœ‘ β†’ ( 1 β€˜π‘ƒ) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ))
103102adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ( 1 β€˜π‘ƒ) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ))
104 fvco3 6990 . . . . . . 7 (((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ) ∧ ( 1 β€˜π‘ƒ) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ)) β†’ ((((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ)) ∘ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ))β€˜( 1 β€˜π‘ƒ)) = (((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ))β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))))
10593, 103, 104syl2anc 582 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ)) ∘ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ))β€˜( 1 β€˜π‘ƒ)) = (((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ))β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))))
10693, 103ffvelcdmd 7088 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ))
107 fvco3 6990 . . . . . . . 8 (((π‘Žβ€˜π‘ƒ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟢((1st β€˜πΉ)β€˜π‘ƒ) ∧ ((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)) β†’ (((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ))β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))) = ((π΄β€˜π‘ƒ)β€˜((π‘Žβ€˜π‘ƒ)β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)))))
10873, 106, 107syl2anc 582 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ))β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))) = ((π΄β€˜π‘ƒ)β€˜((π‘Žβ€˜π‘ƒ)β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)))))
10917adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝐢 ∈ Cat)
11028adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝑋 ∈ 𝐡)
111 eqid 2725 . . . . . . . . . . . 12 (compβ€˜πΆ) = (compβ€˜πΆ)
11230adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋))
113100adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ( 1 β€˜π‘ƒ) ∈ (𝑃(Hom β€˜πΆ)𝑃))
11416, 9, 109, 37, 13, 110, 111, 37, 112, 113yon2 18252 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)) = (𝐾(βŸ¨π‘ƒ, π‘ƒβŸ©(compβ€˜πΆ)𝑋)( 1 β€˜π‘ƒ)))
1159, 13, 99, 109, 37, 111, 110, 112catrid 17658 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (𝐾(βŸ¨π‘ƒ, π‘ƒβŸ©(compβ€˜πΆ)𝑋)( 1 β€˜π‘ƒ)) = 𝐾)
116114, 115eqtrd 2765 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)) = 𝐾)
117116fveq2d 6894 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘ƒ)β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))) = ((π‘Žβ€˜π‘ƒ)β€˜πΎ))
118 eqid 2725 . . . . . . . . . . . . . . 15 (Hom β€˜π‘‚) = (Hom β€˜π‘‚)
11910, 118, 70, 47, 28, 27funcf2 17848 . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃):(𝑋(Hom β€˜π‘‚)𝑃)⟢(((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)))
12013, 8oppchom 17690 . . . . . . . . . . . . . . 15 (𝑋(Hom β€˜π‘‚)𝑃) = (𝑃(Hom β€˜πΆ)𝑋)
12130, 120eleqtrrdi 2836 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐾 ∈ (𝑋(Hom β€˜π‘‚)𝑃))
122119, 121ffvelcdmd 7088 . . . . . . . . . . . . 13 (πœ‘ β†’ ((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)))
12351, 28ffvelcdmd 7088 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) ∈ π‘ˆ)
12418, 22, 70, 123, 52elsetchom 18064 . . . . . . . . . . . . 13 (πœ‘ β†’ (((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)) ↔ ((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)))
125122, 124mpbid 231 . . . . . . . . . . . 12 (πœ‘ β†’ ((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ))
126125adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ))
1279, 13, 99, 17, 28catidcl 17656 . . . . . . . . . . . . 13 (πœ‘ β†’ ( 1 β€˜π‘‹) ∈ (𝑋(Hom β€˜πΆ)𝑋))
12816, 9, 17, 28, 13, 28yon11 18250 . . . . . . . . . . . . 13 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) = (𝑋(Hom β€˜πΆ)𝑋))
129127, 128eleqtrrd 2828 . . . . . . . . . . . 12 (πœ‘ β†’ ( 1 β€˜π‘‹) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹))
130129adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ( 1 β€˜π‘‹) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹))
131 fvco3 6990 . . . . . . . . . . 11 ((((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)⟢((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ) ∧ ( 1 β€˜π‘‹) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)) β†’ (((π‘Žβ€˜π‘ƒ) ∘ ((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ))β€˜( 1 β€˜π‘‹)) = ((π‘Žβ€˜π‘ƒ)β€˜(((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ)β€˜( 1 β€˜π‘‹))))
132126, 130, 131syl2anc 582 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((π‘Žβ€˜π‘ƒ) ∘ ((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ))β€˜( 1 β€˜π‘‹)) = ((π‘Žβ€˜π‘ƒ)β€˜(((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ)β€˜( 1 β€˜π‘‹))))
133121adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝐾 ∈ (𝑋(Hom β€˜π‘‚)𝑃))
1347, 69, 10, 118, 11, 110, 37, 133nati 17939 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘ƒ)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹), ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΉ)β€˜π‘ƒ))((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ)) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹), ((1st β€˜πΉ)β€˜π‘‹)⟩(compβ€˜π‘†)((1st β€˜πΉ)β€˜π‘ƒ))(π‘Žβ€˜π‘‹)))
135123adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) ∈ π‘ˆ)
13618, 40, 11, 135, 53, 61, 126, 73setcco 18066 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘ƒ)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹), ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΉ)β€˜π‘ƒ))((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ)) = ((π‘Žβ€˜π‘ƒ) ∘ ((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ)))
13759, 28ffvelcdmd 7088 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((1st β€˜πΉ)β€˜π‘‹) ∈ π‘ˆ)
138137adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((1st β€˜πΉ)β€˜π‘‹) ∈ π‘ˆ)
1397, 69, 10, 70, 110natcl 17937 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (π‘Žβ€˜π‘‹) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜πΉ)β€˜π‘‹)))
14018, 40, 70, 135, 138elsetchom 18064 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘‹) ∈ (((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜πΉ)β€˜π‘‹)) ↔ (π‘Žβ€˜π‘‹):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)⟢((1st β€˜πΉ)β€˜π‘‹)))
141139, 140mpbid 231 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (π‘Žβ€˜π‘‹):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)⟢((1st β€˜πΉ)β€˜π‘‹))
14210, 118, 70, 56, 28, 27funcf2 17848 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (𝑋(2nd β€˜πΉ)𝑃):(𝑋(Hom β€˜π‘‚)𝑃)⟢(((1st β€˜πΉ)β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜πΉ)β€˜π‘ƒ)))
143142, 121ffvelcdmd 7088 . . . . . . . . . . . . . . 15 (πœ‘ β†’ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ) ∈ (((1st β€˜πΉ)β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜πΉ)β€˜π‘ƒ)))
14418, 22, 70, 137, 60elsetchom 18064 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ) ∈ (((1st β€˜πΉ)β€˜π‘‹)(Hom β€˜π‘†)((1st β€˜πΉ)β€˜π‘ƒ)) ↔ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ):((1st β€˜πΉ)β€˜π‘‹)⟢((1st β€˜πΉ)β€˜π‘ƒ)))
145143, 144mpbid 231 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ):((1st β€˜πΉ)β€˜π‘‹)⟢((1st β€˜πΉ)β€˜π‘ƒ))
146145adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ):((1st β€˜πΉ)β€˜π‘‹)⟢((1st β€˜πΉ)β€˜π‘ƒ))
14718, 40, 11, 135, 138, 61, 141, 146setcco 18066 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)(⟨((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹), ((1st β€˜πΉ)β€˜π‘‹)⟩(compβ€˜π‘†)((1st β€˜πΉ)β€˜π‘ƒ))(π‘Žβ€˜π‘‹)) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ) ∘ (π‘Žβ€˜π‘‹)))
148134, 136, 1473eqtr3d 2773 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘ƒ) ∘ ((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ)) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ) ∘ (π‘Žβ€˜π‘‹)))
149148fveq1d 6892 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((π‘Žβ€˜π‘ƒ) ∘ ((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ))β€˜( 1 β€˜π‘‹)) = ((((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ) ∘ (π‘Žβ€˜π‘‹))β€˜( 1 β€˜π‘‹)))
150127adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ( 1 β€˜π‘‹) ∈ (𝑋(Hom β€˜πΆ)𝑋))
15116, 9, 109, 110, 13, 110, 111, 37, 112, 150yon12 18251 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ)β€˜( 1 β€˜π‘‹)) = (( 1 β€˜π‘‹)(βŸ¨π‘ƒ, π‘‹βŸ©(compβ€˜πΆ)𝑋)𝐾))
1529, 13, 99, 109, 37, 111, 110, 112catlid 17657 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (( 1 β€˜π‘‹)(βŸ¨π‘ƒ, π‘‹βŸ©(compβ€˜πΆ)𝑋)𝐾) = 𝐾)
153151, 152eqtrd 2765 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ)β€˜( 1 β€˜π‘‹)) = 𝐾)
154153fveq2d 6894 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘ƒ)β€˜(((𝑋(2nd β€˜((1st β€˜π‘Œ)β€˜π‘‹))𝑃)β€˜πΎ)β€˜( 1 β€˜π‘‹))) = ((π‘Žβ€˜π‘ƒ)β€˜πΎ))
155132, 149, 1543eqtr3d 2773 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ) ∘ (π‘Žβ€˜π‘‹))β€˜( 1 β€˜π‘‹)) = ((π‘Žβ€˜π‘ƒ)β€˜πΎ))
156 fvco3 6990 . . . . . . . . . 10 (((π‘Žβ€˜π‘‹):((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)⟢((1st β€˜πΉ)β€˜π‘‹) ∧ ( 1 β€˜π‘‹) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹)) β†’ ((((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ) ∘ (π‘Žβ€˜π‘‹))β€˜( 1 β€˜π‘‹)) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))
157141, 130, 156syl2anc 582 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ) ∘ (π‘Žβ€˜π‘‹))β€˜( 1 β€˜π‘‹)) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))
158117, 155, 1573eqtr2d 2771 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π‘Žβ€˜π‘ƒ)β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))) = (((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))
159158fveq2d 6894 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((π΄β€˜π‘ƒ)β€˜((π‘Žβ€˜π‘ƒ)β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)))) = ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)))))
160108, 159eqtrd 2765 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ (((π΄β€˜π‘ƒ) ∘ (π‘Žβ€˜π‘ƒ))β€˜((((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))) = ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)))))
16198, 105, 1603eqtrd 2769 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)) = ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)))))
162161mpteq2dva 5244 . . . 4 (πœ‘ β†’ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)π‘Ž)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))))
1635, 162eqtrid 2777 . . 3 (πœ‘ β†’ (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))))
164 eqid 2725 . . . . . . . . . . 11 (𝑄 Γ—c 𝑂) = (𝑄 Γ—c 𝑂)
165164, 43, 10xpcbas 18163 . . . . . . . . . 10 ((𝑂 Func 𝑆) Γ— 𝐡) = (Baseβ€˜(𝑄 Γ—c 𝑂))
166 eqid 2725 . . . . . . . . . 10 (Hom β€˜(𝑄 Γ—c 𝑂)) = (Hom β€˜(𝑄 Γ—c 𝑂))
167 eqid 2725 . . . . . . . . . 10 (Hom β€˜π‘‡) = (Hom β€˜π‘‡)
168 relfunc 17842 . . . . . . . . . . 11 Rel ((𝑄 Γ—c 𝑂) Func 𝑇)
169 yoneda.t . . . . . . . . . . . . 13 𝑇 = (SetCatβ€˜π‘‰)
170 yoneda.h . . . . . . . . . . . . 13 𝐻 = (HomFβ€˜π‘„)
171 yoneda.r . . . . . . . . . . . . 13 𝑅 = ((𝑄 Γ—c 𝑂) FuncCat 𝑇)
172 yoneda.e . . . . . . . . . . . . 13 𝐸 = (𝑂 evalF 𝑆)
173 yoneda.z . . . . . . . . . . . . 13 𝑍 = (𝐻 ∘func ((⟨(1st β€˜π‘Œ), tpos (2nd β€˜π‘Œ)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17416, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20yonedalem1 18258 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)))
175174simpld 493 . . . . . . . . . . 11 (πœ‘ β†’ 𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
176 1st2ndbr 8040 . . . . . . . . . . 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‘))
177168, 175, 176sylancr 585 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜π‘)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜π‘))
17854, 28opelxpd 5712 . . . . . . . . . 10 (πœ‘ β†’ ⟨𝐹, π‘‹βŸ© ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
17962, 27opelxpd 5712 . . . . . . . . . 10 (πœ‘ β†’ ⟨𝐺, π‘ƒβŸ© ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
180165, 166, 167, 177, 178, 179funcf2 17848 . . . . . . . . 9 (πœ‘ β†’ (⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©):(⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©)⟢(((1st β€˜π‘)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜π‘)β€˜βŸ¨πΊ, π‘ƒβŸ©)))
181164, 43, 10, 14, 118, 54, 28, 62, 27, 166xpchom2 18171 . . . . . . . . . . 11 (πœ‘ β†’ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©) = ((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑋(Hom β€˜π‘‚)𝑃)))
182120xpeq2i 5700 . . . . . . . . . . 11 ((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑋(Hom β€˜π‘‚)𝑃)) = ((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑃(Hom β€˜πΆ)𝑋))
183181, 182eqtrdi 2781 . . . . . . . . . 10 (πœ‘ β†’ (⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©) = ((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑃(Hom β€˜πΆ)𝑋)))
184 df-ov 7416 . . . . . . . . . . . . 13 (𝐹(1st β€˜π‘)𝑋) = ((1st β€˜π‘)β€˜βŸ¨πΉ, π‘‹βŸ©)
185 df-ov 7416 . . . . . . . . . . . . 13 (𝐺(1st β€˜π‘)𝑃) = ((1st β€˜π‘)β€˜βŸ¨πΊ, π‘ƒβŸ©)
186184, 185oveq12i 7425 . . . . . . . . . . . 12 ((𝐹(1st β€˜π‘)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜π‘)𝑃)) = (((1st β€˜π‘)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜π‘)β€˜βŸ¨πΊ, π‘ƒβŸ©))
187186eqcomi 2734 . . . . . . . . . . 11 (((1st β€˜π‘)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜π‘)β€˜βŸ¨πΊ, π‘ƒβŸ©)) = ((𝐹(1st β€˜π‘)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜π‘)𝑃))
188187a1i 11 . . . . . . . . . 10 (πœ‘ β†’ (((1st β€˜π‘)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜π‘)β€˜βŸ¨πΊ, π‘ƒβŸ©)) = ((𝐹(1st β€˜π‘)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜π‘)𝑃)))
189183, 188feq23d 6712 . . . . . . . . 9 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©):(⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©)⟢(((1st β€˜π‘)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜π‘)β€˜βŸ¨πΊ, π‘ƒβŸ©)) ↔ (⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©):((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑃(Hom β€˜πΆ)𝑋))⟢((𝐹(1st β€˜π‘)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜π‘)𝑃))))
190180, 189mpbid 231 . . . . . . . 8 (πœ‘ β†’ (⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©):((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑃(Hom β€˜πΆ)𝑋))⟢((𝐹(1st β€˜π‘)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜π‘)𝑃)))
191190, 34, 30fovcdmd 7587 . . . . . . 7 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾) ∈ ((𝐹(1st β€˜π‘)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜π‘)𝑃)))
192 eqid 2725 . . . . . . . . . . 11 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
193165, 192, 177funcf1 17846 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜π‘):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
194193, 54, 28fovcdmd 7587 . . . . . . . . 9 (πœ‘ β†’ (𝐹(1st β€˜π‘)𝑋) ∈ (Baseβ€˜π‘‡))
195169, 19setcbas 18061 . . . . . . . . 9 (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘‡))
196194, 195eleqtrrd 2828 . . . . . . . 8 (πœ‘ β†’ (𝐹(1st β€˜π‘)𝑋) ∈ 𝑉)
197193, 62, 27fovcdmd 7587 . . . . . . . . 9 (πœ‘ β†’ (𝐺(1st β€˜π‘)𝑃) ∈ (Baseβ€˜π‘‡))
198197, 195eleqtrrd 2828 . . . . . . . 8 (πœ‘ β†’ (𝐺(1st β€˜π‘)𝑃) ∈ 𝑉)
199169, 19, 167, 196, 198elsetchom 18064 . . . . . . 7 (πœ‘ β†’ ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾) ∈ ((𝐹(1st β€˜π‘)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜π‘)𝑃)) ↔ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾):(𝐹(1st β€˜π‘)𝑋)⟢(𝐺(1st β€˜π‘)𝑃)))
200191, 199mpbid 231 . . . . . 6 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾):(𝐹(1st β€˜π‘)𝑋)⟢(𝐺(1st β€˜π‘)𝑃))
20116, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28, 62, 27, 34, 30yonedalem22 18264 . . . . . . . 8 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾) = (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(2nd β€˜π»)⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)𝐴))
2028oppccat 17698 . . . . . . . . . . 11 (𝐢 ∈ Cat β†’ 𝑂 ∈ Cat)
20317, 202syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑂 ∈ Cat)
20418setccat 18068 . . . . . . . . . . 11 (π‘ˆ ∈ V β†’ 𝑆 ∈ Cat)
20522, 204syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑆 ∈ Cat)
2066, 203, 205fuccat 17956 . . . . . . . . 9 (πœ‘ β†’ 𝑄 ∈ Cat)
207170, 206, 43, 14, 45, 54, 82, 62, 12, 31, 34hof2val 18242 . . . . . . . 8 (πœ‘ β†’ (((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(2nd β€˜π»)⟨((1st β€˜π‘Œ)β€˜π‘ƒ), 𝐺⟩)𝐴) = (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))))
208201, 207eqtrd 2765 . . . . . . 7 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾) = (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))))
20916, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28yonedalem21 18259 . . . . . . 7 (πœ‘ β†’ (𝐹(1st β€˜π‘)𝑋) = (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹))
21016, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 62, 27yonedalem21 18259 . . . . . . 7 (πœ‘ β†’ (𝐺(1st β€˜π‘)𝑃) = (((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)𝐺))
211208, 209, 210feq123d 6706 . . . . . 6 (πœ‘ β†’ ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾):(𝐹(1st β€˜π‘)𝑋)⟢(𝐺(1st β€˜π‘)𝑃) ↔ (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))):(((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)⟢(((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)𝐺)))
212200, 211mpbid 231 . . . . 5 (πœ‘ β†’ (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))):(((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)⟢(((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)𝐺))
213 eqid 2725 . . . . . 6 (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))) = (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)))
214213fmpt 7113 . . . . 5 (βˆ€π‘ ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)) ∈ (((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)𝐺) ↔ (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))):(((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)⟢(((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)𝐺))
215212, 214sylibr 233 . . . 4 (πœ‘ β†’ βˆ€π‘ ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)) ∈ (((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)𝐺))
216 yonedalem3.m . . . . . 6 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‘₯))))
21716, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 62, 27, 216yonedalem3a 18260 . . . . 5 (πœ‘ β†’ ((𝐺𝑀𝑃) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)𝐺) ↦ ((π‘Žβ€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))) ∧ (𝐺𝑀𝑃):(𝐺(1st β€˜π‘)𝑃)⟢(𝐺(1st β€˜πΈ)𝑃)))
218217simpld 493 . . . 4 (πœ‘ β†’ (𝐺𝑀𝑃) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)𝐺) ↦ ((π‘Žβ€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))))
219 fveq1 6889 . . . . 5 (π‘Ž = ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)) β†’ (π‘Žβ€˜π‘ƒ) = (((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ))
220219fveq1d 6892 . . . 4 (π‘Ž = ((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ)) β†’ ((π‘Žβ€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)) = ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ)))
221215, 208, 218, 220fmptcof 7133 . . 3 (πœ‘ β†’ ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾)) = (𝑏 ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((((𝐴(⟨((1st β€˜π‘Œ)β€˜π‘‹), 𝐹⟩(compβ€˜π‘„)𝐺)𝑏)(⟨((1st β€˜π‘Œ)β€˜π‘ƒ), ((1st β€˜π‘Œ)β€˜π‘‹)⟩(compβ€˜π‘„)𝐺)((𝑃(2nd β€˜π‘Œ)𝑋)β€˜πΎ))β€˜π‘ƒ)β€˜( 1 β€˜π‘ƒ))))
222 eqid 2725 . . . . . . 7 (⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©) = (⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)
223172, 203, 205, 10, 118, 11, 7, 54, 62, 28, 27, 222, 34, 121evlf2val 18205 . . . . . 6 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾) = ((π΄β€˜π‘ƒ)(⟨((1st β€˜πΉ)β€˜π‘‹), ((1st β€˜πΉ)β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ))((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)))
22418, 22, 11, 137, 60, 67, 145, 77setcco 18066 . . . . . 6 (πœ‘ β†’ ((π΄β€˜π‘ƒ)(⟨((1st β€˜πΉ)β€˜π‘‹), ((1st β€˜πΉ)β€˜π‘ƒ)⟩(compβ€˜π‘†)((1st β€˜πΊ)β€˜π‘ƒ))((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)) = ((π΄β€˜π‘ƒ) ∘ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)))
225223, 224eqtrd 2765 . . . . 5 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾) = ((π΄β€˜π‘ƒ) ∘ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)))
226225coeq1d 5859 . . . 4 (πœ‘ β†’ ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾) ∘ (𝐹𝑀𝑋)) = (((π΄β€˜π‘ƒ) ∘ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)) ∘ (𝐹𝑀𝑋)))
22716, 9, 99, 8, 18, 169, 6, 170, 171, 172, 173, 17, 19, 23, 20, 54, 28, 216yonedalem3a 18260 . . . . . . . 8 (πœ‘ β†’ ((𝐹𝑀𝑋) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))) ∧ (𝐹𝑀𝑋):(𝐹(1st β€˜π‘)𝑋)⟢(𝐹(1st β€˜πΈ)𝑋)))
228227simprd 494 . . . . . . 7 (πœ‘ β†’ (𝐹𝑀𝑋):(𝐹(1st β€˜π‘)𝑋)⟢(𝐹(1st β€˜πΈ)𝑋))
229227simpld 493 . . . . . . . 8 (πœ‘ β†’ (𝐹𝑀𝑋) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))
230172, 203, 205, 10, 54, 28evlf1 18206 . . . . . . . 8 (πœ‘ β†’ (𝐹(1st β€˜πΈ)𝑋) = ((1st β€˜πΉ)β€˜π‘‹))
231229, 209, 230feq123d 6706 . . . . . . 7 (πœ‘ β†’ ((𝐹𝑀𝑋):(𝐹(1st β€˜π‘)𝑋)⟢(𝐹(1st β€˜πΈ)𝑋) ↔ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))):(((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)⟢((1st β€˜πΉ)β€˜π‘‹)))
232228, 231mpbid 231 . . . . . 6 (πœ‘ β†’ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))):(((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)⟢((1st β€˜πΉ)β€˜π‘‹))
233 eqid 2725 . . . . . . 7 (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)))
234233fmpt 7113 . . . . . 6 (βˆ€π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)) ∈ ((1st β€˜πΉ)β€˜π‘‹) ↔ (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))):(((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)⟢((1st β€˜πΉ)β€˜π‘‹))
235232, 234sylibr 233 . . . . 5 (πœ‘ β†’ βˆ€π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)) ∈ ((1st β€˜πΉ)β€˜π‘‹))
236 fcompt 7136 . . . . . 6 (((π΄β€˜π‘ƒ):((1st β€˜πΉ)β€˜π‘ƒ)⟢((1st β€˜πΊ)β€˜π‘ƒ) ∧ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ):((1st β€˜πΉ)β€˜π‘‹)⟢((1st β€˜πΉ)β€˜π‘ƒ)) β†’ ((π΄β€˜π‘ƒ) ∘ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)) = (𝑦 ∈ ((1st β€˜πΉ)β€˜π‘‹) ↦ ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜π‘¦))))
23777, 145, 236syl2anc 582 . . . . 5 (πœ‘ β†’ ((π΄β€˜π‘ƒ) ∘ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)) = (𝑦 ∈ ((1st β€˜πΉ)β€˜π‘‹) ↦ ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜π‘¦))))
238 2fveq3 6895 . . . . 5 (𝑦 = ((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)) β†’ ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜π‘¦)) = ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹)))))
239235, 229, 237, 238fmptcof 7133 . . . 4 (πœ‘ β†’ (((π΄β€˜π‘ƒ) ∘ ((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)) ∘ (𝐹𝑀𝑋)) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))))
240226, 239eqtrd 2765 . . 3 (πœ‘ β†’ ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾) ∘ (𝐹𝑀𝑋)) = (π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹) ↦ ((π΄β€˜π‘ƒ)β€˜(((𝑋(2nd β€˜πΉ)𝑃)β€˜πΎ)β€˜((π‘Žβ€˜π‘‹)β€˜( 1 β€˜π‘‹))))))
241163, 221, 2403eqtr4d 2775 . 2 (πœ‘ β†’ ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾)) = ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾) ∘ (𝐹𝑀𝑋)))
242 eqid 2725 . . 3 (compβ€˜π‘‡) = (compβ€˜π‘‡)
243174simprd 494 . . . . . . 7 (πœ‘ β†’ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
244 1st2ndbr 8040 . . . . . . 7 ((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))
245168, 243, 244sylancr 585 . . . . . 6 (πœ‘ β†’ (1st β€˜πΈ)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜πΈ))
246165, 192, 245funcf1 17846 . . . . 5 (πœ‘ β†’ (1st β€˜πΈ):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
247246, 62, 27fovcdmd 7587 . . . 4 (πœ‘ β†’ (𝐺(1st β€˜πΈ)𝑃) ∈ (Baseβ€˜π‘‡))
248247, 195eleqtrrd 2828 . . 3 (πœ‘ β†’ (𝐺(1st β€˜πΈ)𝑃) ∈ 𝑉)
249217simprd 494 . . 3 (πœ‘ β†’ (𝐺𝑀𝑃):(𝐺(1st β€˜π‘)𝑃)⟢(𝐺(1st β€˜πΈ)𝑃))
250169, 19, 242, 196, 198, 248, 200, 249setcco 18066 . 2 (πœ‘ β†’ ((𝐺𝑀𝑃)(⟨(𝐹(1st β€˜π‘)𝑋), (𝐺(1st β€˜π‘)𝑃)⟩(compβ€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃))(𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾)) = ((𝐺𝑀𝑃) ∘ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾)))
251246, 54, 28fovcdmd 7587 . . . 4 (πœ‘ β†’ (𝐹(1st β€˜πΈ)𝑋) ∈ (Baseβ€˜π‘‡))
252251, 195eleqtrrd 2828 . . 3 (πœ‘ β†’ (𝐹(1st β€˜πΈ)𝑋) ∈ 𝑉)
253165, 166, 167, 245, 178, 179funcf2 17848 . . . . . 6 (πœ‘ β†’ (⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©):(⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©)⟢(((1st β€˜πΈ)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜βŸ¨πΊ, π‘ƒβŸ©)))
254 df-ov 7416 . . . . . . . . . 10 (𝐹(1st β€˜πΈ)𝑋) = ((1st β€˜πΈ)β€˜βŸ¨πΉ, π‘‹βŸ©)
255 df-ov 7416 . . . . . . . . . 10 (𝐺(1st β€˜πΈ)𝑃) = ((1st β€˜πΈ)β€˜βŸ¨πΊ, π‘ƒβŸ©)
256254, 255oveq12i 7425 . . . . . . . . 9 ((𝐹(1st β€˜πΈ)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃)) = (((1st β€˜πΈ)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜βŸ¨πΊ, π‘ƒβŸ©))
257256eqcomi 2734 . . . . . . . 8 (((1st β€˜πΈ)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜βŸ¨πΊ, π‘ƒβŸ©)) = ((𝐹(1st β€˜πΈ)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃))
258257a1i 11 . . . . . . 7 (πœ‘ β†’ (((1st β€˜πΈ)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜βŸ¨πΊ, π‘ƒβŸ©)) = ((𝐹(1st β€˜πΈ)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃)))
259183, 258feq23d 6712 . . . . . 6 (πœ‘ β†’ ((⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©):(⟨𝐹, π‘‹βŸ©(Hom β€˜(𝑄 Γ—c 𝑂))⟨𝐺, π‘ƒβŸ©)⟢(((1st β€˜πΈ)β€˜βŸ¨πΉ, π‘‹βŸ©)(Hom β€˜π‘‡)((1st β€˜πΈ)β€˜βŸ¨πΊ, π‘ƒβŸ©)) ↔ (⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©):((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑃(Hom β€˜πΆ)𝑋))⟢((𝐹(1st β€˜πΈ)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃))))
260253, 259mpbid 231 . . . . 5 (πœ‘ β†’ (⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©):((𝐹(𝑂 Nat 𝑆)𝐺) Γ— (𝑃(Hom β€˜πΆ)𝑋))⟢((𝐹(1st β€˜πΈ)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃)))
261260, 34, 30fovcdmd 7587 . . . 4 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾) ∈ ((𝐹(1st β€˜πΈ)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃)))
262169, 19, 167, 252, 248elsetchom 18064 . . . 4 (πœ‘ β†’ ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾) ∈ ((𝐹(1st β€˜πΈ)𝑋)(Hom β€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃)) ↔ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾):(𝐹(1st β€˜πΈ)𝑋)⟢(𝐺(1st β€˜πΈ)𝑃)))
263261, 262mpbid 231 . . 3 (πœ‘ β†’ (𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾):(𝐹(1st β€˜πΈ)𝑋)⟢(𝐺(1st β€˜πΈ)𝑃))
264169, 19, 242, 196, 252, 248, 228, 263setcco 18066 . 2 (πœ‘ β†’ ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾)(⟨(𝐹(1st β€˜π‘)𝑋), (𝐹(1st β€˜πΈ)𝑋)⟩(compβ€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃))(𝐹𝑀𝑋)) = ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾) ∘ (𝐹𝑀𝑋)))
265241, 250, 2643eqtr4d 2775 1 (πœ‘ β†’ ((𝐺𝑀𝑃)(⟨(𝐹(1st β€˜π‘)𝑋), (𝐺(1st β€˜π‘)𝑃)⟩(compβ€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃))(𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜π‘)⟨𝐺, π‘ƒβŸ©)𝐾)) = ((𝐴(⟨𝐹, π‘‹βŸ©(2nd β€˜πΈ)⟨𝐺, π‘ƒβŸ©)𝐾)(⟨(𝐹(1st β€˜π‘)𝑋), (𝐹(1st β€˜πΈ)𝑋)⟩(compβ€˜π‘‡)(𝐺(1st β€˜πΈ)𝑃))(𝐹𝑀𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  Vcvv 3463   βˆͺ cun 3939   βŠ† wss 3941  βŸ¨cop 4631   class class class wbr 5144   ↦ cmpt 5227   Γ— cxp 5671  ran crn 5674   ∘ ccom 5677  Rel wrel 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7413   ∈ cmpo 7415  1st c1st 7985  2nd c2nd 7986  tpos ctpos 8224  Basecbs 17174  Hom chom 17238  compcco 17239  Catccat 17638  Idccid 17639  Homf chomf 17640  oppCatcoppc 17685   Func cfunc 17834   ∘func ccofu 17836   Nat cnat 17925   FuncCat cfuc 17926  SetCatcsetc 18058   Γ—c cxpc 18153   1stF c1stf 18154   2ndF c2ndf 18155   ⟨,⟩F cprf 18156   evalF cevlf 18195  HomFchof 18234  Yoncyon 18235
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 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
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 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-hom 17251  df-cco 17252  df-cat 17642  df-cid 17643  df-homf 17644  df-comf 17645  df-oppc 17686  df-ssc 17787  df-resc 17788  df-subc 17789  df-func 17838  df-cofu 17840  df-nat 17927  df-fuc 17928  df-setc 18059  df-xpc 18157  df-1stf 18158  df-2ndf 18159  df-prf 18160  df-evlf 18199  df-curf 18200  df-hof 18236  df-yon 18237
This theorem is referenced by:  yonedalem3  18266
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