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Theorem invfuc 18023
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 2736 . . . . . . . . . 10 (Base‘𝐷) = (Base‘𝐷)
4 fucinv.j . . . . . . . . . 10 𝐽 = (Inv‘𝐷)
5 fuciso.f . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
6 funcrcl 17909 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
75, 6syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
87simprd 495 . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
98adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
10 fuciso.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐶)
11 relfunc 17908 . . . . . . . . . . . . 13 Rel (𝐶 Func 𝐷)
12 1st2ndbr 8068 . . . . . . . . . . . . 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 17912 . . . . . . . . . . 11 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
1514ffvelcdmda 7103 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
16 fuciso.g . . . . . . . . . . . . 13 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
17 1st2ndbr 8068 . . . . . . . . . . . . 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 17912 . . . . . . . . . . 11 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
2019ffvelcdmda 7103 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
21 eqid 2736 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
223, 4, 9, 15, 20, 21invss 17806 . . . . . . . . 9 ((𝜑𝑥𝐵) → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ⊆ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)) × (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
2322ssbrd 5185 . . . . . . . 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 5733 . . . . . . . 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 496 . . . . . . 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 3145 . . . . 5 (𝜑 → ∀𝑥𝐵 𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2910fvexi 6919 . . . . . 6 𝐵 ∈ V
30 mptelixpg 8976 . . . . . 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 234 . . . 4 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
33 fveq2 6905 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑦))
34 fveq2 6905 . . . . . 6 (𝑥 = 𝑦 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑦))
3533, 34oveq12d 7450 . . . . 5 (𝑥 = 𝑦 → (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
3635cbvixpv 8956 . . . 4 X𝑥𝐵 (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) = X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦))
3732, 36eleqtrdi 2850 . . 3 (𝜑 → (𝑥𝐵𝑋) ∈ X𝑦𝐵 (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
38 simpr2 1195 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
39 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐵) → 𝑥𝐵)
40 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (𝑥𝐵𝑋) = (𝑥𝐵𝑋)
4140fvmpt2 7026 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵𝑋 ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
4239, 27, 41syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐵) → ((𝑥𝐵𝑋)‘𝑥) = 𝑋)
432, 42breqtrrd 5170 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
4443ralrimiva 3145 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
4544adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥))
46 nfcv 2904 . . . . . . . . . . . . . . . 16 𝑥(𝑈𝑧)
47 nfcv 2904 . . . . . . . . . . . . . . . 16 𝑥(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))
48 nffvmpt1 6916 . . . . . . . . . . . . . . . 16 𝑥((𝑥𝐵𝑋)‘𝑧)
4946, 47, 48nfbr 5189 . . . . . . . . . . . . . . 15 𝑥(𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)
50 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑈𝑥) = (𝑈𝑧))
51 fveq2 6905 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑧))
52 fveq2 6905 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑧))
5351, 52oveq12d 7450 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧)))
54 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑧))
5550, 53, 54breq123d 5156 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑧)(((1st𝐹)‘𝑧)𝐽((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧)))
5649, 55rspc 3609 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐷 ∈ Cat)
5914adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹):𝐵⟶(Base‘𝐷))
6059, 38ffvelcdmd 7104 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑧) ∈ (Base‘𝐷))
6119adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺):𝐵⟶(Base‘𝐷))
6261, 38ffvelcdmd 7104 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑧) ∈ (Base‘𝐷))
63 eqid 2736 . . . . . . . . . . . . . 14 (Sect‘𝐷) = (Sect‘𝐷)
643, 4, 58, 60, 62, 63isinv 17805 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 494 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧)(((1st𝐹)‘𝑧)(Sect‘𝐷)((1st𝐺)‘𝑧))((𝑥𝐵𝑋)‘𝑧))
67 eqid 2736 . . . . . . . . . . . 12 (comp‘𝐷) = (comp‘𝐷)
68 eqid 2736 . . . . . . . . . . . 12 (Id‘𝐷) = (Id‘𝐷)
693, 21, 67, 68, 63, 58, 60, 62issect 17798 . . . . . . . . . . 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 232 . . . . . . . . . 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 1144 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑥𝐵𝑋)‘𝑧)(⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩(comp‘𝐷)((1st𝐹)‘𝑧))(𝑈𝑧)) = ((Id‘𝐷)‘((1st𝐹)‘𝑧)))
7271oveq1d 7447 . . . . . . . 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 1194 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
7459, 73ffvelcdmd 7104 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
75 eqid 2736 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
7613adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
7710, 75, 21, 76, 73, 38funcf2 17914 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
78 simpr3 1196 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))
7977, 78ffvelcdmd 7104 . . . . . . . . 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 17727 . . . . . . . 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 2777 . . . . . . 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 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (𝐹𝑁𝐺))
8482, 83nat1st2nd 18000 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑈 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
8582, 84, 10, 21, 38natcl 18002 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑧) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
8670simp2d 1143 . . . . . . . 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 17730 . . . . . . 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 18004 . . . . . . . 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 7448 . . . . . . 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 2780 . . . . . 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 7447 . . . . 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, 73ffvelcdmd 7104 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
93 nfcv 2904 . . . . . . . . . . . . 13 𝑥(𝑈𝑦)
94 nfcv 2904 . . . . . . . . . . . . 13 𝑥(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))
95 nffvmpt1 6916 . . . . . . . . . . . . 13 𝑥((𝑥𝐵𝑋)‘𝑦)
9693, 94, 95nfbr 5189 . . . . . . . . . . . 12 𝑥(𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)
97 fveq2 6905 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑈𝑥) = (𝑈𝑦))
9834, 33oveq12d 7450 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
99 fveq2 6905 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥𝐵𝑋)‘𝑥) = ((𝑥𝐵𝑋)‘𝑦))
10097, 98, 99breq123d 5156 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)))
10196, 100rspc 3609 . . . . . . . . . . 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 17805 . . . . . . . . . 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 232 . . . . . . . . 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 495 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑦)(((1st𝐺)‘𝑦)(Sect‘𝐷)((1st𝐹)‘𝑦))(𝑈𝑦))
1063, 21, 67, 68, 63, 58, 92, 74issect 17798 . . . . . . . 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 232 . . . . . . 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 1142 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥𝐵𝑋)‘𝑦) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
109107simp2d 1143 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑦) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑦)))
11018adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
11110, 75, 21, 110, 73, 38funcf2 17914 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐺)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐺)‘𝑦)(Hom ‘𝐷)((1st𝐺)‘𝑧)))
112111, 78ffvelcdmd 7104 . . . . . . 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 17729 . . . . . 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 17730 . . . . 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 18002 . . . . . . . 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 17730 . . . . . . 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 1144 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑧𝐵𝑓 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑈𝑦)(⟨((1st𝐺)‘𝑦), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦)) = ((Id‘𝐷)‘((1st𝐺)‘𝑦)))
118117oveq2d 7448 . . . . . . 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 17728 . . . . . . 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 2780 . . . . . 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 7448 . . . . 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 2781 . . . 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 3204 . . 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 17997 . . 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 713 . 2 (𝜑 → (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹))
126 nfv 1913 . . . 4 𝑦(𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥)
127126, 96, 100cbvralw 3305 . . 3 (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))((𝑥𝐵𝑋)‘𝑥) ↔ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))
12844, 127sylib 218 . 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 18022 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ (𝑥𝐵𝑋) ∈ (𝐺𝑁𝐹) ∧ ∀𝑦𝐵 (𝑈𝑦)(((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))((𝑥𝐵𝑋)‘𝑦))))
1321, 125, 128, 131mpbir3and 1342 1 (𝜑𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cop 4631   class class class wbr 5142  cmpt 5224   × cxp 5682  Rel wrel 5689  wf 6556  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  Xcixp 8938  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709  Sectcsect 17789  Invcinv 17790   Func cfunc 17900   Nat cnat 17990   FuncCat cfuc 17991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-cat 17712  df-cid 17713  df-sect 17792  df-inv 17793  df-func 17904  df-nat 17992  df-fuc 17993
This theorem is referenced by:  fuciso  18024  yonedainv  18327
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