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Theorem yonedalem3b 18256
Description: Lemma for yoneda 18260. (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 7422 . . . . . . . 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 7429 . . . . . . 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 6893 . . . . . 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 6893 . . . . 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 5255 . . . 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 2727 . . . . . . . . 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 17684 . . . . . . . . 9 𝐡 = (Baseβ€˜π‘‚)
11 eqid 2727 . . . . . . . . 9 (compβ€˜π‘†) = (compβ€˜π‘†)
12 eqid 2727 . . . . . . . . 9 (compβ€˜π‘„) = (compβ€˜π‘„)
13 eqid 2727 . . . . . . . . . . . 12 (Hom β€˜πΆ) = (Hom β€˜πΆ)
146, 7fuchom 17937 . . . . . . . . . . . 12 (𝑂 Nat 𝑆) = (Hom β€˜π‘„)
15 relfunc 17833 . . . . . . . . . . . . 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 4184 . . . . . . . . . . . . . . 15 (πœ‘ β†’ π‘ˆ βŠ† 𝑉)
2219, 21ssexd 5318 . . . . . . . . . . . . . 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 18239 . . . . . . . . . . . . 13 (πœ‘ β†’ π‘Œ ∈ (𝐢 Func 𝑄))
25 1st2ndbr 8038 . . . . . . . . . . . . 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 586 . . . . . . . . . . . 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 17839 . . . . . . . . . . 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 7089 . . . . . . . . . 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 480 . . . . . . . . 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 484 . . . . . . . . . 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 480 . . . . . . . . . 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 17941 . . . . . . . . 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 480 . . . . . . . . 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 17940 . . . . . . . 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 17940 . . . . . . . . . 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 480 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ π‘ˆ ∈ V)
41 eqid 2727 . . . . . . . . . . . . . . 15 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
42 relfunc 17833 . . . . . . . . . . . . . . . 16 Rel (𝑂 Func 𝑆)
436fucbas 17936 . . . . . . . . . . . . . . . . . 18 (𝑂 Func 𝑆) = (Baseβ€˜π‘„)
449, 43, 26funcf1 17837 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (1st β€˜π‘Œ):𝐡⟢(𝑂 Func 𝑆))
4544, 28ffvelcdmd 7089 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ ((1st β€˜π‘Œ)β€˜π‘‹) ∈ (𝑂 Func 𝑆))
46 1st2ndbr 8038 . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . 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 17837 . . . . . . . . . . . . . 14 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)):𝐡⟢(Baseβ€˜π‘†))
4918, 22setcbas 18052 . . . . . . . . . . . . . . 15 (πœ‘ β†’ π‘ˆ = (Baseβ€˜π‘†))
5049feq3d 6703 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 13 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘‹)):π΅βŸΆπ‘ˆ)
5251, 27ffvelcdmd 7089 . . . . . . . . . . . 12 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘ƒ) ∈ π‘ˆ)
5352adantr 480 . . . . . . . . . . 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 8038 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜πΉ)(𝑂 Func 𝑆)(2nd β€˜πΉ))
5642, 54, 55sylancr 586 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (1st β€˜πΉ)(𝑂 Func 𝑆)(2nd β€˜πΉ))
5710, 41, 56funcf1 17837 . . . . . . . . . . . . . 14 (πœ‘ β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π‘†))
5849feq3d 6703 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((1st β€˜πΉ):π΅βŸΆπ‘ˆ ↔ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π‘†)))
5957, 58mpbird 257 . . . . . . . . . . . . 13 (πœ‘ β†’ (1st β€˜πΉ):π΅βŸΆπ‘ˆ)
6059, 27ffvelcdmd 7089 . . . . . . . . . . . 12 (πœ‘ β†’ ((1st β€˜πΉ)β€˜π‘ƒ) ∈ π‘ˆ)
6160adantr 480 . . . . . . . . . . 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 8038 . . . . . . . . . . . . . . . 16 ((Rel (𝑂 Func 𝑆) ∧ 𝐺 ∈ (𝑂 Func 𝑆)) β†’ (1st β€˜πΊ)(𝑂 Func 𝑆)(2nd β€˜πΊ))
6442, 62, 63sylancr 586 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (1st β€˜πΊ)(𝑂 Func 𝑆)(2nd β€˜πΊ))
6510, 41, 64funcf1 17837 . . . . . . . . . . . . . 14 (πœ‘ β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π‘†))
6665, 27ffvelcdmd 7089 . . . . . . . . . . . . 13 (πœ‘ β†’ ((1st β€˜πΊ)β€˜π‘ƒ) ∈ (Baseβ€˜π‘†))
6766, 49eleqtrrd 2831 . . . . . . . . . . . 12 (πœ‘ β†’ ((1st β€˜πΊ)β€˜π‘ƒ) ∈ π‘ˆ)
6867adantr 480 . . . . . . . . . . 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 17926 . . . . . . . . . . . . 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 2727 . . . . . . . . . . . . 13 (Hom β€˜π‘†) = (Hom β€˜π‘†)
717, 69, 10, 70, 37natcl 17928 . . . . . . . . . . . 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 18055 . . . . . . . . . . . 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 17926 . . . . . . . . . . . . . 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 17928 . . . . . . . . . . . . 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 18055 . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 18057 . . . . . . . . . 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 2767 . . . . . . . . 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 7429 . . . . . . . 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 7089 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((1st β€˜π‘Œ)β€˜π‘ƒ) ∈ (𝑂 Func 𝑆))
83 1st2ndbr 8038 . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . 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 17837 . . . . . . . . . . . 12 (πœ‘ β†’ (1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ)):𝐡⟢(Baseβ€˜π‘†))
8685, 27ffvelcdmd 7089 . . . . . . . . . . 11 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ) ∈ (Baseβ€˜π‘†))
8786, 49eleqtrrd 2831 . . . . . . . . . 10 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ) ∈ π‘ˆ)
8887adantr 480 . . . . . . . . 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 17926 . . . . . . . . . . . 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 17928 . . . . . . . . . . 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 18055 . . . . . . . . . . 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 480 . . . . . . . . 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 583 . . . . . . . . 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 18057 . . . . . . . 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 2771 . . . . . . 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 6893 . . . . . 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 17647 . . . . . . . . 9 (πœ‘ β†’ ( 1 β€˜π‘ƒ) ∈ (𝑃(Hom β€˜πΆ)𝑃))
10116, 9, 17, 27, 13, 27yon11 18241 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ) = (𝑃(Hom β€˜πΆ)𝑃))
102100, 101eleqtrrd 2831 . . . . . . . 8 (πœ‘ β†’ ( 1 β€˜π‘ƒ) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘ƒ))β€˜π‘ƒ))
103102adantr 480 . . . . . . 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 6991 . . . . . . 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 583 . . . . . 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 7089 . . . . . . . 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 6991 . . . . . . . 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 583 . . . . . . 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 480 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝐢 ∈ Cat)
11028adantr 480 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝑋 ∈ 𝐡)
111 eqid 2727 . . . . . . . . . . . 12 (compβ€˜πΆ) = (compβ€˜πΆ)
11230adantr 480 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (((1st β€˜π‘Œ)β€˜π‘‹)(𝑂 Nat 𝑆)𝐹)) β†’ 𝐾 ∈ (𝑃(Hom β€˜πΆ)𝑋))
113100adantr 480 . . . . . . . . . . . 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 18243 . . . . . . . . . . 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 17649 . . . . . . . . . . 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 2767 . . . . . . . . . 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 6895 . . . . . . . . 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 2727 . . . . . . . . . . . . . . 15 (Hom β€˜π‘‚) = (Hom β€˜π‘‚)
11910, 118, 70, 47, 28, 27funcf2 17839 . . . . . . . . . . . . . 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 17681 . . . . . . . . . . . . . . 15 (𝑋(Hom β€˜π‘‚)𝑃) = (𝑃(Hom β€˜πΆ)𝑋)
12130, 120eleqtrrdi 2839 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐾 ∈ (𝑋(Hom β€˜π‘‚)𝑃))
122119, 121ffvelcdmd 7089 . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) ∈ π‘ˆ)
12418, 22, 70, 123, 52elsetchom 18055 . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 17647 . . . . . . . . . . . . 13 (πœ‘ β†’ ( 1 β€˜π‘‹) ∈ (𝑋(Hom β€˜πΆ)𝑋))
12816, 9, 17, 28, 13, 28yon11 18241 . . . . . . . . . . . . 13 (πœ‘ β†’ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹) = (𝑋(Hom β€˜πΆ)𝑋))
129127, 128eleqtrrd 2831 . . . . . . . . . . . 12 (πœ‘ β†’ ( 1 β€˜π‘‹) ∈ ((1st β€˜((1st β€˜π‘Œ)β€˜π‘‹))β€˜π‘‹))
130129adantr 480 . . . . . . . . . . 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 6991 . . . . . . . . . . 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 583 . . . . . . . . . 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 480 . . . . . . . . . . . . 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 17930 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 18057 . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((1st β€˜πΉ)β€˜π‘‹) ∈ π‘ˆ)
138137adantr 480 . . . . . . . . . . . . 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 17928 . . . . . . . . . . . . . 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 18055 . . . . . . . . . . . . . 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 17839 . . . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . . 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 18055 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 18057 . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 6893 . . . . . . . . . 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 480 . . . . . . . . . . . . 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 18242 . . . . . . . . . . . 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 17648 . . . . . . . . . . . 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 2767 . . . . . . . . . . 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 6895 . . . . . . . . . 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 2775 . . . . . . . . 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 6991 . . . . . . . . . 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 583 . . . . . . . . 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 2773 . . . . . . . 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 6895 . . . . . . 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 2767 . . . . . 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 2771 . . . . 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 5242 . . . 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 2779 . . 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 2727 . . . . . . . . . . 11 (𝑄 Γ—c 𝑂) = (𝑄 Γ—c 𝑂)
165164, 43, 10xpcbas 18154 . . . . . . . . . 10 ((𝑂 Func 𝑆) Γ— 𝐡) = (Baseβ€˜(𝑄 Γ—c 𝑂))
166 eqid 2727 . . . . . . . . . 10 (Hom β€˜(𝑄 Γ—c 𝑂)) = (Hom β€˜(𝑄 Γ—c 𝑂))
167 eqid 2727 . . . . . . . . . 10 (Hom β€˜π‘‡) = (Hom β€˜π‘‡)
168 relfunc 17833 . . . . . . . . . . 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 18249 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇)))
175174simpld 494 . . . . . . . . . . 11 (πœ‘ β†’ 𝑍 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
176 1st2ndbr 8038 . . . . . . . . . . 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 586 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜π‘)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜π‘))
17854, 28opelxpd 5711 . . . . . . . . . 10 (πœ‘ β†’ ⟨𝐹, π‘‹βŸ© ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
17962, 27opelxpd 5711 . . . . . . . . . 10 (πœ‘ β†’ ⟨𝐺, π‘ƒβŸ© ∈ ((𝑂 Func 𝑆) Γ— 𝐡))
180165, 166, 167, 177, 178, 179funcf2 17839 . . . . . . . . 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 18162 . . . . . . . . . . 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 5699 . . . . . . . . . . 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 2783 . . . . . . . . . 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 7417 . . . . . . . . . . . . 13 (𝐹(1st β€˜π‘)𝑋) = ((1st β€˜π‘)β€˜βŸ¨πΉ, π‘‹βŸ©)
185 df-ov 7417 . . . . . . . . . . . . 13 (𝐺(1st β€˜π‘)𝑃) = ((1st β€˜π‘)β€˜βŸ¨πΊ, π‘ƒβŸ©)
186184, 185oveq12i 7426 . . . . . . . . . . . 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 2736 . . . . . . . . . . 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 6711 . . . . . . . . 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 7585 . . . . . . 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 2727 . . . . . . . . . . 11 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
193165, 192, 177funcf1 17837 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜π‘):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
194193, 54, 28fovcdmd 7585 . . . . . . . . 9 (πœ‘ β†’ (𝐹(1st β€˜π‘)𝑋) ∈ (Baseβ€˜π‘‡))
195169, 19setcbas 18052 . . . . . . . . 9 (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘‡))
196194, 195eleqtrrd 2831 . . . . . . . 8 (πœ‘ β†’ (𝐹(1st β€˜π‘)𝑋) ∈ 𝑉)
197193, 62, 27fovcdmd 7585 . . . . . . . . 9 (πœ‘ β†’ (𝐺(1st β€˜π‘)𝑃) ∈ (Baseβ€˜π‘‡))
198197, 195eleqtrrd 2831 . . . . . . . 8 (πœ‘ β†’ (𝐺(1st β€˜π‘)𝑃) ∈ 𝑉)
199169, 19, 167, 196, 198elsetchom 18055 . . . . . . 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 18255 . . . . . . . 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 17689 . . . . . . . . . . 11 (𝐢 ∈ Cat β†’ 𝑂 ∈ Cat)
20317, 202syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑂 ∈ Cat)
20418setccat 18059 . . . . . . . . . . 11 (π‘ˆ ∈ V β†’ 𝑆 ∈ Cat)
20522, 204syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑆 ∈ Cat)
2066, 203, 205fuccat 17947 . . . . . . . . 9 (πœ‘ β†’ 𝑄 ∈ Cat)
207170, 206, 43, 14, 45, 54, 82, 62, 12, 31, 34hof2val 18233 . . . . . . . 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 2767 . . . . . . 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 18250 . . . . . . 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 18250 . . . . . . 7 (πœ‘ β†’ (𝐺(1st β€˜π‘)𝑃) = (((1st β€˜π‘Œ)β€˜π‘ƒ)(𝑂 Nat 𝑆)𝐺))
211208, 209, 210feq123d 6705 . . . . . 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 2727 . . . . . 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 7114 . . . . 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 18251 . . . . 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 494 . . . 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 6890 . . . . 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 6893 . . . 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 2727 . . . . . . 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 18196 . . . . . 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 18057 . . . . . 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 2767 . . . . 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 5858 . . . 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 18251 . . . . . . . 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 495 . . . . . . 7 (πœ‘ β†’ (𝐹𝑀𝑋):(𝐹(1st β€˜π‘)𝑋)⟢(𝐹(1st β€˜πΈ)𝑋))
229227simpld 494 . . . . . . . 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 18197 . . . . . . . 8 (πœ‘ β†’ (𝐹(1st β€˜πΈ)𝑋) = ((1st β€˜πΉ)β€˜π‘‹))
231229, 209, 230feq123d 6705 . . . . . . 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 2727 . . . . . . 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 7114 . . . . . 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 583 . . . . 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 6896 . . . . 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 2767 . . 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 2777 . 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 2727 . . 3 (compβ€˜π‘‡) = (compβ€˜π‘‡)
243174simprd 495 . . . . . . 7 (πœ‘ β†’ 𝐸 ∈ ((𝑄 Γ—c 𝑂) Func 𝑇))
244 1st2ndbr 8038 . . . . . . 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 586 . . . . . 6 (πœ‘ β†’ (1st β€˜πΈ)((𝑄 Γ—c 𝑂) Func 𝑇)(2nd β€˜πΈ))
246165, 192, 245funcf1 17837 . . . . 5 (πœ‘ β†’ (1st β€˜πΈ):((𝑂 Func 𝑆) Γ— 𝐡)⟢(Baseβ€˜π‘‡))
247246, 62, 27fovcdmd 7585 . . . 4 (πœ‘ β†’ (𝐺(1st β€˜πΈ)𝑃) ∈ (Baseβ€˜π‘‡))
248247, 195eleqtrrd 2831 . . 3 (πœ‘ β†’ (𝐺(1st β€˜πΈ)𝑃) ∈ 𝑉)
249217simprd 495 . . 3 (πœ‘ β†’ (𝐺𝑀𝑃):(𝐺(1st β€˜π‘)𝑃)⟢(𝐺(1st β€˜πΈ)𝑃))
250169, 19, 242, 196, 198, 248, 200, 249setcco 18057 . 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 7585 . . . 4 (πœ‘ β†’ (𝐹(1st β€˜πΈ)𝑋) ∈ (Baseβ€˜π‘‡))
252251, 195eleqtrrd 2831 . . 3 (πœ‘ β†’ (𝐹(1st β€˜πΈ)𝑋) ∈ 𝑉)
253165, 166, 167, 245, 178, 179funcf2 17839 . . . . . 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 7417 . . . . . . . . . 10 (𝐹(1st β€˜πΈ)𝑋) = ((1st β€˜πΈ)β€˜βŸ¨πΉ, π‘‹βŸ©)
255 df-ov 7417 . . . . . . . . . 10 (𝐺(1st β€˜πΈ)𝑃) = ((1st β€˜πΈ)β€˜βŸ¨πΊ, π‘ƒβŸ©)
256254, 255oveq12i 7426 . . . . . . . . 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 2736 . . . . . . . 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 6711 . . . . . 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 7585 . . . 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 18055 . . . 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 18057 . 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 2777 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 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  Vcvv 3469   βˆͺ cun 3942   βŠ† wss 3944  βŸ¨cop 4630   class class class wbr 5142   ↦ cmpt 5225   Γ— cxp 5670  ran crn 5673   ∘ ccom 5676  Rel wrel 5677  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  1st c1st 7983  2nd c2nd 7984  tpos ctpos 8222  Basecbs 17165  Hom chom 17229  compcco 17230  Catccat 17629  Idccid 17630  Homf chomf 17631  oppCatcoppc 17676   Func cfunc 17825   ∘func ccofu 17827   Nat cnat 17916   FuncCat cfuc 17917  SetCatcsetc 18049   Γ—c cxpc 18144   1stF c1stf 18145   2ndF c2ndf 18146   ⟨,⟩F cprf 18147   evalF cevlf 18186  HomFchof 18225  Yoncyon 18226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  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 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-tpos 8223  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-pm 8837  df-ixp 8906  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12489  df-z 12575  df-dec 12694  df-uz 12839  df-fz 13503  df-struct 17101  df-sets 17118  df-slot 17136  df-ndx 17148  df-base 17166  df-ress 17195  df-hom 17242  df-cco 17243  df-cat 17633  df-cid 17634  df-homf 17635  df-comf 17636  df-oppc 17677  df-ssc 17778  df-resc 17779  df-subc 17780  df-func 17829  df-cofu 17831  df-nat 17918  df-fuc 17919  df-setc 18050  df-xpc 18148  df-1stf 18149  df-2ndf 18150  df-prf 18151  df-evlf 18190  df-curf 18191  df-hof 18227  df-yon 18228
This theorem is referenced by:  yonedalem3  18257
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