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Theorem yonffthlem 16854
Description: Lemma for yonffth 16856. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y 𝑌 = (Yon‘𝐶)
yoneda.b 𝐵 = (Base‘𝐶)
yoneda.1 1 = (Id‘𝐶)
yoneda.o 𝑂 = (oppCat‘𝐶)
yoneda.s 𝑆 = (SetCat‘𝑈)
yoneda.t 𝑇 = (SetCat‘𝑉)
yoneda.q 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h 𝐻 = (HomF𝑄)
yoneda.r 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (𝜑𝐶 ∈ Cat)
yoneda.w (𝜑𝑉𝑊)
yoneda.u (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoneda.v (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
yoneda.m 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
yonedainv.i 𝐼 = (Inv‘𝑅)
yonedainv.n 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
Assertion
Ref Expression
yonffthlem (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
Distinct variable groups:   𝑓,𝑎,𝑔,𝑥,𝑦, 1   𝑢,𝑎,𝑔,𝑦,𝐶,𝑓,𝑥   𝐸,𝑎,𝑓,𝑔,𝑢,𝑦   𝐵,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑁,𝑎   𝑂,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑆,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑔,𝑀,𝑢,𝑦   𝑄,𝑎,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝜑,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑢,𝑅   𝑌,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑍,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦
Allowed substitution hints:   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   1 (𝑢)   𝐸(𝑥)   𝐻(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝐼(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑁(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑊(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)

Proof of Theorem yonffthlem
Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 16454 . . 3 Rel (𝐶 Func 𝑄)
2 yoneda.y . . . 4 𝑌 = (Yon‘𝐶)
3 yoneda.c . . . 4 (𝜑𝐶 ∈ Cat)
4 yoneda.o . . . 4 𝑂 = (oppCat‘𝐶)
5 yoneda.s . . . 4 𝑆 = (SetCat‘𝑈)
6 yoneda.q . . . 4 𝑄 = (𝑂 FuncCat 𝑆)
7 yoneda.w . . . . 5 (𝜑𝑉𝑊)
8 yoneda.v . . . . . 6 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
98unssbd 3774 . . . . 5 (𝜑𝑈𝑉)
107, 9ssexd 4770 . . . 4 (𝜑𝑈 ∈ V)
11 yoneda.u . . . 4 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
122, 3, 4, 5, 6, 10, 11yoncl 16834 . . 3 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
13 1st2nd 7166 . . 3 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
141, 12, 13sylancr 694 . 2 (𝜑𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
15 1st2ndbr 7169 . . . . 5 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
161, 12, 15sylancr 694 . . . 4 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
17 yoneda.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐶)
186fucbas 16552 . . . . . . . . . . . . 13 (𝑂 Func 𝑆) = (Base‘𝑄)
1917, 18, 16funcf1 16458 . . . . . . . . . . . 12 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
2019adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
21 simprr 795 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑤𝐵)
2220, 21ffvelrnd 6321 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
23 simprl 793 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑧𝐵)
24 opelxpi 5113 . . . . . . . . . 10 ((((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆) ∧ 𝑧𝐵) → ⟨((1st𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
2522, 23, 24syl2anc 692 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ⟨((1st𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
26 yoneda.r . . . . . . . . . . . . . 14 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
2726fucbas 16552 . . . . . . . . . . . . 13 ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
28 yonedainv.i . . . . . . . . . . . . 13 𝐼 = (Inv‘𝑅)
29 yoneda.1 . . . . . . . . . . . . . . . . . 18 1 = (Id‘𝐶)
30 yoneda.t . . . . . . . . . . . . . . . . . 18 𝑇 = (SetCat‘𝑉)
31 yoneda.h . . . . . . . . . . . . . . . . . 18 𝐻 = (HomF𝑄)
32 yoneda.e . . . . . . . . . . . . . . . . . 18 𝐸 = (𝑂 evalF 𝑆)
33 yoneda.z . . . . . . . . . . . . . . . . . 18 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
342, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 3, 7, 11, 8yonedalem1 16844 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
3534simpld 475 . . . . . . . . . . . . . . . 16 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
36 funcrcl 16455 . . . . . . . . . . . . . . . 16 (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
3735, 36syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
3837simpld 475 . . . . . . . . . . . . . 14 (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
3937simprd 479 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ Cat)
4026, 38, 39fuccat 16562 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ Cat)
4134simprd 479 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
42 eqid 2621 . . . . . . . . . . . . 13 (Iso‘𝑅) = (Iso‘𝑅)
43 yoneda.m . . . . . . . . . . . . . 14 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
44 yonedainv.n . . . . . . . . . . . . . 14 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
452, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 3, 7, 11, 8, 43, 28, 44yonedainv 16853 . . . . . . . . . . . . 13 (𝜑𝑀(𝑍𝐼𝐸)𝑁)
4627, 28, 40, 35, 41, 42, 45inviso2 16359 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (𝐸(Iso‘𝑅)𝑍))
47 eqid 2621 . . . . . . . . . . . . . 14 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
484, 17oppcbas 16310 . . . . . . . . . . . . . 14 𝐵 = (Base‘𝑂)
4947, 18, 48xpcbas 16750 . . . . . . . . . . . . 13 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
50 eqid 2621 . . . . . . . . . . . . 13 ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
51 eqid 2621 . . . . . . . . . . . . 13 (Iso‘𝑇) = (Iso‘𝑇)
5226, 49, 50, 41, 35, 42, 51fuciso 16567 . . . . . . . . . . . 12 (𝜑 → (𝑁 ∈ (𝐸(Iso‘𝑅)𝑍) ↔ (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)))))
5346, 52mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣))))
5453simprd 479 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)))
5554adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)))
56 fveq2 6153 . . . . . . . . . . . 12 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → (𝑁𝑣) = (𝑁‘⟨((1st𝑌)‘𝑤), 𝑧⟩))
57 df-ov 6613 . . . . . . . . . . . 12 (((1st𝑌)‘𝑤)𝑁𝑧) = (𝑁‘⟨((1st𝑌)‘𝑤), 𝑧⟩)
5856, 57syl6eqr 2673 . . . . . . . . . . 11 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → (𝑁𝑣) = (((1st𝑌)‘𝑤)𝑁𝑧))
59 fveq2 6153 . . . . . . . . . . . . 13 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝐸)‘𝑣) = ((1st𝐸)‘⟨((1st𝑌)‘𝑤), 𝑧⟩))
60 df-ov 6613 . . . . . . . . . . . . 13 (((1st𝑌)‘𝑤)(1st𝐸)𝑧) = ((1st𝐸)‘⟨((1st𝑌)‘𝑤), 𝑧⟩)
6159, 60syl6eqr 2673 . . . . . . . . . . . 12 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝐸)‘𝑣) = (((1st𝑌)‘𝑤)(1st𝐸)𝑧))
62 fveq2 6153 . . . . . . . . . . . . 13 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝑍)‘𝑣) = ((1st𝑍)‘⟨((1st𝑌)‘𝑤), 𝑧⟩))
63 df-ov 6613 . . . . . . . . . . . . 13 (((1st𝑌)‘𝑤)(1st𝑍)𝑧) = ((1st𝑍)‘⟨((1st𝑌)‘𝑤), 𝑧⟩)
6462, 63syl6eqr 2673 . . . . . . . . . . . 12 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝑍)‘𝑣) = (((1st𝑌)‘𝑤)(1st𝑍)𝑧))
6561, 64oveq12d 6628 . . . . . . . . . . 11 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)) = ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧)))
6658, 65eleq12d 2692 . . . . . . . . . 10 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)) ↔ (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧))))
6766rspcv 3294 . . . . . . . . 9 (⟨((1st𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵) → (∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)) → (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧))))
6825, 55, 67sylc 65 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧)))
694oppccat 16314 . . . . . . . . . . . . 13 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
703, 69syl 17 . . . . . . . . . . . 12 (𝜑𝑂 ∈ Cat)
7170adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑂 ∈ Cat)
725setccat 16667 . . . . . . . . . . . . 13 (𝑈 ∈ V → 𝑆 ∈ Cat)
7310, 72syl 17 . . . . . . . . . . . 12 (𝜑𝑆 ∈ Cat)
7473adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑆 ∈ Cat)
7532, 71, 74, 48, 22, 23evlf1 16792 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)(1st𝐸)𝑧) = ((1st ‘((1st𝑌)‘𝑤))‘𝑧))
763adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝐶 ∈ Cat)
77 eqid 2621 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
782, 17, 76, 21, 77, 23yon11 16836 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
7975, 78eqtrd 2655 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)(1st𝐸)𝑧) = (𝑧(Hom ‘𝐶)𝑤))
807adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑉𝑊)
8111adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ran (Homf𝐶) ⊆ 𝑈)
828adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
832, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 76, 80, 81, 82, 22, 23yonedalem21 16845 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)(1st𝑍)𝑧) = (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
8479, 83oveq12d 6628 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧)) = ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
8568, 84eleqtrd 2700 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
869adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑈𝑉)
87 eqid 2621 . . . . . . . . . . . . 13 (Base‘𝑆) = (Base‘𝑆)
88 relfunc 16454 . . . . . . . . . . . . . 14 Rel (𝑂 Func 𝑆)
89 1st2ndbr 7169 . . . . . . . . . . . . . 14 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
9088, 22, 89sylancr 694 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
9148, 87, 90funcf1 16458 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
9291, 23ffvelrnd 6321 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆))
935, 10setcbas 16660 . . . . . . . . . . . 12 (𝜑𝑈 = (Base‘𝑆))
9493adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑈 = (Base‘𝑆))
9592, 94eleqtrrd 2701 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
9678, 95eqeltrrd 2699 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈)
9786, 96sseldd 3588 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉)
98 eqid 2621 . . . . . . . . . 10 (Homf𝑄) = (Homf𝑄)
99 eqid 2621 . . . . . . . . . . 11 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
1006, 99fuchom 16553 . . . . . . . . . 10 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
10120, 23ffvelrnd 6321 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
10298, 18, 100, 101, 22homfval 16284 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) = (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
1038unssad 3773 . . . . . . . . . . 11 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
104103adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ran (Homf𝑄) ⊆ 𝑉)
10598, 18homffn 16285 . . . . . . . . . . . 12 (Homf𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆))
106105a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (Homf𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)))
107 fnovrn 6769 . . . . . . . . . . 11 (((Homf𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)) ∧ ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) ∈ ran (Homf𝑄))
108106, 101, 22, 107syl3anc 1323 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) ∈ ran (Homf𝑄))
109104, 108sseldd 3588 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) ∈ 𝑉)
110102, 109eqeltrrd 2699 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)) ∈ 𝑉)
11130, 80, 97, 110, 51setciso 16673 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))) ↔ (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
11285, 111mpbid 222 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
11376adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat)
114113adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝐶 ∈ Cat)
11523adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧𝐵)
116115adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑧𝐵)
117 simpr 477 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑦𝐵)
1182, 17, 114, 116, 77, 117yon11 16836 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑧))‘𝑦) = (𝑦(Hom ‘𝐶)𝑧))
119118eqcomd 2627 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (𝑦(Hom ‘𝐶)𝑧) = ((1st ‘((1st𝑌)‘𝑧))‘𝑦))
120114adantr 481 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝐶 ∈ Cat)
12121ad3antrrr 765 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑤𝐵)
122116adantr 481 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧𝐵)
123 eqid 2621 . . . . . . . . . . . . . . 15 (comp‘𝐶) = (comp‘𝐶)
124117adantr 481 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦𝐵)
125 simpr 477 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
126 simpllr 798 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ∈ (𝑧(Hom ‘𝐶)𝑤))
1272, 17, 120, 121, 77, 122, 123, 124, 125, 126yon12 16837 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘) = ((⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
1282, 17, 120, 122, 77, 121, 123, 124, 126, 125yon2 16838 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔) = ((⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
129127, 128eqtr4d 2658 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘) = ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔))
130119, 129mpteq12dva 4697 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘)) = (𝑔 ∈ ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔)))
13116adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
13217, 77, 100, 131, 23, 21funcf2 16460 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
133132ffvelrnda 6320 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) ∈ (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
13499, 133nat1st2nd 16543 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) ∈ (⟨(1st ‘((1st𝑌)‘𝑧)), (2nd ‘((1st𝑌)‘𝑧))⟩(𝑂 Nat 𝑆)⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩))
135134adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((𝑧(2nd𝑌)𝑤)‘) ∈ (⟨(1st ‘((1st𝑌)‘𝑧)), (2nd ‘((1st𝑌)‘𝑧))⟩(𝑂 Nat 𝑆)⟨(1st ‘((1st𝑌)‘𝑤)), (2nd ‘((1st𝑌)‘𝑤))⟩))
136 eqid 2621 . . . . . . . . . . . . . . 15 (Hom ‘𝑆) = (Hom ‘𝑆)
13799, 135, 48, 136, 117natcl 16545 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (((𝑧(2nd𝑌)𝑤)‘)‘𝑦) ∈ (((1st ‘((1st𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑦)))
13810adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑈 ∈ V)
139138ad2antrr 761 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑈 ∈ V)
14019ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
141140, 115ffvelrnd 6321 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
142 1st2ndbr 7169 . . . . . . . . . . . . . . . . . . 19 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑧)))
14388, 141, 142sylancr 694 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑧)))
14448, 87, 143funcf1 16458 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑧)):𝐵⟶(Base‘𝑆))
145144ffvelrnda 6320 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆))
14694ad2antrr 761 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑈 = (Base‘𝑆))
147145, 146eleqtrrd 2701 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ∈ 𝑈)
14891adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
149148ffvelrnda 6320 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆))
150149, 146eleqtrrd 2701 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑦) ∈ 𝑈)
1515, 139, 136, 147, 150elsetchom 16663 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦) ∈ (((1st ‘((1st𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑦)) ↔ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦):((1st ‘((1st𝑌)‘𝑧))‘𝑦)⟶((1st ‘((1st𝑌)‘𝑤))‘𝑦)))
152137, 151mpbid 222 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (((𝑧(2nd𝑌)𝑤)‘)‘𝑦):((1st ‘((1st𝑌)‘𝑧))‘𝑦)⟶((1st ‘((1st𝑌)‘𝑤))‘𝑦))
153152feqmptd 6211 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (((𝑧(2nd𝑌)𝑤)‘)‘𝑦) = (𝑔 ∈ ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔)))
154130, 153eqtr4d 2658 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘)) = (((𝑧(2nd𝑌)𝑤)‘)‘𝑦))
155154mpteq2dva 4709 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘))) = (𝑦𝐵 ↦ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦)))
15680adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉𝑊)
15781adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
15882adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
15922adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
16078eleq2d 2684 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ( ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↔ ∈ (𝑧(Hom ‘𝐶)𝑤)))
161160biimpar 502 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧))
1622, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 113, 156, 157, 158, 159, 115, 44, 161yonedalem4a 16847 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st𝑌)‘𝑤)𝑁𝑧)‘) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘))))
16399, 134, 48natfn 16546 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) Fn 𝐵)
164 dffn5 6203 . . . . . . . . . . 11 (((𝑧(2nd𝑌)𝑤)‘) Fn 𝐵 ↔ ((𝑧(2nd𝑌)𝑤)‘) = (𝑦𝐵 ↦ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦)))
165163, 164sylib 208 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) = (𝑦𝐵 ↦ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦)))
166155, 162, 1653eqtr4d 2665 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st𝑌)‘𝑤)𝑁𝑧)‘) = ((𝑧(2nd𝑌)𝑤)‘))
167166mpteq2dva 4709 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st𝑌)‘𝑤)𝑁𝑧)‘)) = ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd𝑌)𝑤)‘)))
168 f1of 6099 . . . . . . . . . 10 ((((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)) → (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
169112, 168syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
170169feqmptd 6211 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) = ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st𝑌)‘𝑤)𝑁𝑧)‘)))
171132feqmptd 6211 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(2nd𝑌)𝑤) = ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd𝑌)𝑤)‘)))
172167, 170, 1713eqtr4d 2665 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd𝑌)𝑤))
173 f1oeq1 6089 . . . . . . 7 ((((1st𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd𝑌)𝑤) → ((((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)) ↔ (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
174172, 173syl 17 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)) ↔ (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
175112, 174mpbid 222 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
176175ralrimivva 2966 . . . 4 (𝜑 → ∀𝑧𝐵𝑤𝐵 (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
17717, 77, 100isffth2 16508 . . . 4 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ((1st𝑌)(𝐶 Func 𝑄)(2nd𝑌) ∧ ∀𝑧𝐵𝑤𝐵 (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
17816, 176, 177sylanbrc 697 . . 3 (𝜑 → (1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌))
179 df-br 4619 . . 3 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
180178, 179sylib 208 . 2 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
18114, 180eqeltrd 2698 1 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  cun 3557  cin 3558  wss 3559  cop 4159   class class class wbr 4618  cmpt 4678   × cxp 5077  ran crn 5080  Rel wrel 5084   Fn wfn 5847  wf 5848  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cmpt2 6612  1st c1st 7118  2nd c2nd 7119  tpos ctpos 7303  Basecbs 15792  Hom chom 15884  compcco 15885  Catccat 16257  Idccid 16258  Homf chomf 16259  oppCatcoppc 16303  Invcinv 16337  Isociso 16338   Func cfunc 16446  func ccofu 16448   Full cful 16494   Faith cfth 16495   Nat cnat 16533   FuncCat cfuc 16534  SetCatcsetc 16657   ×c cxpc 16740   1stF c1stf 16741   2ndF c2ndf 16742   ⟨,⟩F cprf 16743   evalF cevlf 16781  HomFchof 16820  Yoncyon 16821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-tpos 7304  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-hom 15898  df-cco 15899  df-cat 16261  df-cid 16262  df-homf 16263  df-comf 16264  df-oppc 16304  df-sect 16339  df-inv 16340  df-iso 16341  df-ssc 16402  df-resc 16403  df-subc 16404  df-func 16450  df-cofu 16452  df-full 16496  df-fth 16497  df-nat 16535  df-fuc 16536  df-setc 16658  df-xpc 16744  df-1stf 16745  df-2ndf 16746  df-prf 16747  df-evlf 16785  df-curf 16786  df-hof 16822  df-yon 16823
This theorem is referenced by:  yonffth  16856
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