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Theorem invfuc 17073
Description: If 𝑉(𝑥) is an inverse to 𝑈(𝑥) for each 𝑥, and 𝑈 is a natural transformation, then 𝑉 is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
fuciso.q 𝑄 = (𝐶 FuncCat 𝐷)
fuciso.b 𝐵 = (Base‘𝐶)
fuciso.n 𝑁 = (𝐶 Nat 𝐷)
fuciso.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
fuciso.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
fucinv.i 𝐼 = (Inv‘𝑄)
fucinv.j 𝐽 = (Inv‘𝐷)
invfuc.u (𝜑𝑈 ∈ (𝐹𝑁𝐺))
invfuc.v ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))𝑋)
Assertion
Ref Expression
invfuc (𝜑𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐼   𝑥,𝐹   𝑥,𝐺   𝑥,𝐽   𝑥,𝑁   𝜑,𝑥   𝑥,𝑄   𝑥,𝑈
Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem invfuc
Dummy variables 𝑦 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfuc.u . 2 (𝜑𝑈 ∈ (𝐹𝑁𝐺))
2 invfuc.v . . . . . . . 8 ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))𝑋)
3 eqid 2795 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
4 fucinv.j . . . . . . . . . 10 𝐽 = (Inv‘𝐷)
5 fuciso.f . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
6 funcrcl 16962 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
75, 6syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
87simprd 496 . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
98adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
10 fuciso.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐶)
11 relfunc 16961 . . . . . . . . . . . . 13 Rel (𝐶 Func 𝐷)
12 1st2ndbr 7597 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1311, 5, 12sylancr 587 . . . . . . . . . . . 12 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1410, 3, 13funcf1 16965 . . . . . . . . . . 11 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
1514ffvelrnda 6716 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
16 fuciso.g . . . . . . . . . . . . 13 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
17 1st2ndbr 7597 . . . . . . . . . . . . 13 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1811, 16, 17sylancr 587 . . . . . . . . . . . 12 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1910, 3, 18funcf1 16965 . . . . . . . . . . 11 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
2019ffvelrnda 6716 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
21 eqid 2795 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
223, 4, 9, 15, 20, 21invss 16860 . . . . . . . . 9 ((𝜑𝑥𝐵) → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ⊆ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
2322ssbrd 5005 . . . . . . . 8 ((𝜑𝑥𝐵) → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))𝑋 → (𝑈𝑥)((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))𝑋))
242, 23mpd 15 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑈𝑥)((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))𝑋)
25 brxp 5489 . . . . . . . 8 ((𝑈𝑥)((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))𝑋 ↔ ((𝑈𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) ∧ 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
2625simprbi 497 . . . . . . 7 ((𝑈𝑥)((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))𝑋𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2724, 26syl 17 . . . . . 6 ((𝜑𝑥𝐵) → 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2827ralrimiva 3149 . . . . 5 (𝜑 → ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2910fvexi 6552 . . . . . 6 𝐵 ∈ V
30 mptelixpg 8347 . . . . . 6 (𝐵 ∈ V → ((𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) ↔ ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
3129, 30ax-mp 5 . . . . 5 ((𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) ↔ ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
3228, 31sylibr 235 . . . 4 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
33 fveq2 6538 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑦))
34 fveq2 6538 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑦))
3533, 34oveq12d 7034 . . . . 5 (𝑥 = 𝑦 → (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
3635cbvixpv 8328 . . . 4 X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦))
3732, 36syl6eleq 2893 . . 3 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
38 simpr2 1188 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
39 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐵) → 𝑥𝐵)
40 eqid 2795 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐵𝑋) = (𝑥𝐵𝑋)
4140fvmpt2 6645 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
4239, 27, 41syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐵) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
432, 42breqtrrd 4990 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
4443ralrimiva 3149 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
4544adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
46 nfcv 2949 . . . . . . . . . . . . . . . 16 𝑥(𝑈𝑧)
47 nfcv 2949 . . . . . . . . . . . . . . . 16 𝑥(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))
48 nffvmpt1 6549 . . . . . . . . . . . . . . . 16 𝑥((𝑥𝐵𝑋)‘𝑧)
4946, 47, 48nfbr 5009 . . . . . . . . . . . . . . 15 𝑥(𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)
50 fveq2 6538 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑈𝑥) = (𝑈𝑧))
51 fveq2 6538 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑧))
52 fveq2 6538 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑧))
5351, 52oveq12d 7034 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧)))
54 fveq2 6538 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑧))
5550, 53, 54breq123d 4976 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)))
5649, 55rspc 3553 . . . . . . . . . . . . . 14 (𝑧𝐵 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) → (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)))
5738, 45, 56sylc 65 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧))
588adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
5914adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹):𝐵⟶(Base‘𝐷))
6059, 38ffvelrnd 6717 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑧) ∈ (Base‘𝐷))
6119adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺):𝐵⟶(Base‘𝐷))
6261, 38ffvelrnd 6717 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑧) ∈ (Base‘𝐷))
63 eqid 2795 . . . . . . . . . . . . . 14 (Sect‘𝐷) = (Sect‘𝐷)
643, 4, 58, 60, 62, 63isinv 16859 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧) ↔ ((𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧) ∧ ((𝑥𝐵𝑋)‘𝑧)(((1st𝐺)‘𝑧)(Sect‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧))))
6557, 64mpbid 233 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧) ∧ ((𝑥𝐵𝑋)‘𝑧)(((1st𝐺)‘𝑧)(Sect‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)))
6665simpld 495 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧))
67 eqid 2795 . . . . . . . . . . . 12 (comp‘𝐷) = (comp‘𝐷)
68 eqid 2795 . . . . . . . . . . . 12 (Id‘𝐷) = (Id‘𝐷)
693, 21, 67, 68, 63, 58, 60, 62issect 16852 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧) ↔ ((𝑈𝑧) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝐷)((1st𝐺)‘𝑧)) ∧ ((𝑥𝐵𝑋)‘𝑧) ∈ (((1st𝐺)‘𝑧)(Hom ‘𝐷)((1st𝐹)‘𝑧)) ∧ (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)) = ((Id‘𝐷)‘((1st𝐹)‘𝑧)))))
7066, 69mpbid 233 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝐷)((1st𝐺)‘𝑧)) ∧ ((𝑥𝐵𝑋)‘𝑧) ∈ (((1st𝐺)‘𝑧)(Hom ‘𝐷)((1st𝐹)‘𝑧)) ∧ (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)) = ((Id‘𝐷)‘((1st𝐹)‘𝑧))))
7170simp3d 1137 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)) = ((Id‘𝐷)‘((1st𝐹)‘𝑧)))
7271oveq1d 7031 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)) = (((Id‘𝐷)‘((1st𝐹)‘𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)))
73 simpr1 1187 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
7459, 73ffvelrnd 6717 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
75 eqid 2795 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
7613adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
7710, 75, 21, 76, 73, 38funcf2 16967 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
78 simpr3 1189 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
7977, 78ffvelrnd 6717 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑓) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
803, 21, 68, 58, 74, 67, 60, 79catlid 16783 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((Id‘𝐷)‘((1st𝐹)‘𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)) = ((𝑦(2nd𝐹)𝑧)‘𝑓))
8172, 80eqtr2d 2832 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑓) = ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)))
82 fuciso.n . . . . . . . . 9 𝑁 = (𝐶 Nat 𝐷)
831adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (𝐹𝑁𝐺))
8482, 83nat1st2nd 17050 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
8582, 84, 10, 21, 38natcl 17052 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
8670simp2d 1136 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑧) ∈ (((1st𝐺)‘𝑧)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
873, 21, 67, 58, 74, 60, 62, 79, 85, 60, 86catass 16786 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧))(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑈𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓))))
8882, 84, 10, 75, 67, 73, 38, 78nati 17054 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓)) = (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦)))
8988oveq2d 7032 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑈𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐹)‘𝑧)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑦(2nd𝐹)𝑧)‘𝑓))) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))))
9081, 87, 893eqtrd 2835 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑓) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))))
9190oveq1d 7031 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐹)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)) = ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦)))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)))
9261, 73ffvelrnd 6717 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
93 nfcv 2949 . . . . . . . . . . . . 13 𝑥(𝑈𝑦)
94 nfcv 2949 . . . . . . . . . . . . 13 𝑥(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))
95 nffvmpt1 6549 . . . . . . . . . . . . 13 𝑥((𝑥𝐵𝑋)‘𝑦)
9693, 94, 95nfbr 5009 . . . . . . . . . . . 12 𝑥(𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)
97 fveq2 6538 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑈𝑥) = (𝑈𝑦))
9834, 33oveq12d 7034 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
99 fveq2 6538 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑦))
10097, 98, 99breq123d 4976 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)))
10196, 100rspc 3553 . . . . . . . . . . 11 (𝑦𝐵 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) → (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)))
10273, 45, 101sylc 65 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
1033, 4, 58, 74, 92, 63isinv 16859 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦) ↔ ((𝑈𝑦)(((1st𝐹)‘𝑦)(Sect‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦) ∧ ((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦))))
104102, 103mpbid 233 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑦)(((1st𝐹)‘𝑦)(Sect‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦) ∧ ((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦)))
105104simprd 496 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦))
1063, 21, 67, 68, 63, 58, 92, 74issect 16852 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦) ↔ (((𝑥𝐵𝑋)‘𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ∧ (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)) ∧ ((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)) = ((Id‘𝐷)‘((1st𝐺)‘𝑦)))))
107105, 106mpbid 233 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ∧ (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)) ∧ ((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)) = ((Id‘𝐷)‘((1st𝐺)‘𝑦))))
108107simp1d 1135 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
109107simp2d 1136 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
11018adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
11110, 75, 21, 110, 73, 38funcf2 16967 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐺)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
112111, 78ffvelrnd 6717 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐺)𝑧)‘𝑓) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
1133, 21, 67, 58, 74, 92, 62, 109, 112catcocl 16785 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦)) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
1143, 21, 67, 58, 92, 74, 62, 108, 113, 60, 86catass 16786 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦)))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑦))))
11582, 84, 10, 21, 73natcl 17052 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
1163, 21, 67, 58, 92, 74, 92, 108, 115, 62, 112catass 16786 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑦)) = (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))))
117107simp3d 1137 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)) = ((Id‘𝐷)‘((1st𝐺)‘𝑦)))
118117oveq2d 7032 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))) = (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((Id‘𝐷)‘((1st𝐺)‘𝑦))))
1193, 21, 68, 58, 92, 67, 62, 112catrid 16784 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((Id‘𝐷)‘((1st𝐺)‘𝑦))) = ((𝑦(2nd𝐺)𝑧)‘𝑓))
120116, 118, 1193eqtrd 2835 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑦)) = ((𝑦(2nd𝐺)𝑧)‘𝑓))
121120oveq2d 7032 . . . . 5 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((((𝑦(2nd𝐺)𝑧)‘𝑓)(⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))(𝑈𝑦))(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑦))) = (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐺)𝑧)‘𝑓)))
12291, 114, 1213eqtrrd 2836 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐺)𝑧)‘𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)))
123122ralrimivvva 3159 . . 3 (𝜑 → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐺)𝑧)‘𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)))
12482, 10, 75, 21, 67, 16, 5isnat2 17047 . . 3 (𝜑 → ((𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹) ↔ ((𝑥𝐵𝑋) ∈ X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐺)‘𝑦), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑦(2nd𝐺)𝑧)‘𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑓)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥𝐵𝑋)‘𝑦)))))
12537, 123, 124mpbir2and 709 . 2 (𝜑 → (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹))
126 nfv 1892 . . . 4 𝑦(𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥)
127126, 96, 100cbvral 3399 . . 3 (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
12844, 127sylib 219 . 2 (𝜑 → ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
129 fuciso.q . . 3 𝑄 = (𝐶 FuncCat 𝐷)
130 fucinv.i . . 3 𝐼 = (Inv‘𝑄)
131129, 10, 82, 5, 16, 130, 4fucinv 17072 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹) ∧ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))))
1321, 125, 128, 131mpbir3and 1335 1 (𝜑𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  Vcvv 3437  cop 4478   class class class wbr 4962  cmpt 5041   × cxp 5441  Rel wrel 5448  wf 6221  cfv 6225  (class class class)co 7016  1st c1st 7543  2nd c2nd 7544  Xcixp 8310  Basecbs 16312  Hom chom 16405  compcco 16406  Catccat 16764  Idccid 16765  Sectcsect 16843  Invcinv 16844   Func cfunc 16953   Nat cnat 17040   FuncCat cfuc 17041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-hom 16418  df-cco 16419  df-cat 16768  df-cid 16769  df-sect 16846  df-inv 16847  df-func 16957  df-nat 17042  df-fuc 17043
This theorem is referenced by:  fuciso  17074  yonedainv  17360
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