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Theorem yonffthlem 17129
 Description: Lemma for yonffth 17131. (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 16728 . . 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 3940 . . . . 5 (𝜑𝑈𝑉)
107, 9ssexd 4936 . . . 4 (𝜑𝑈 ∈ V)
11 yoneda.u . . . 4 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
122, 3, 4, 5, 6, 10, 11yoncl 17109 . . 3 (𝜑𝑌 ∈ (𝐶 Func 𝑄))
13 1st2nd 7362 . . 3 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
141, 12, 13sylancr 567 . 2 (𝜑𝑌 = ⟨(1st𝑌), (2nd𝑌)⟩)
15 1st2ndbr 7365 . . . . 5 ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
161, 12, 15sylancr 567 . . . 4 (𝜑 → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
17 fveq2 6332 . . . . . . . . . . 11 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → (𝑁𝑣) = (𝑁‘⟨((1st𝑌)‘𝑤), 𝑧⟩))
18 df-ov 6795 . . . . . . . . . . 11 (((1st𝑌)‘𝑤)𝑁𝑧) = (𝑁‘⟨((1st𝑌)‘𝑤), 𝑧⟩)
1917, 18syl6eqr 2822 . . . . . . . . . 10 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → (𝑁𝑣) = (((1st𝑌)‘𝑤)𝑁𝑧))
20 fveq2 6332 . . . . . . . . . . . 12 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝐸)‘𝑣) = ((1st𝐸)‘⟨((1st𝑌)‘𝑤), 𝑧⟩))
21 df-ov 6795 . . . . . . . . . . . 12 (((1st𝑌)‘𝑤)(1st𝐸)𝑧) = ((1st𝐸)‘⟨((1st𝑌)‘𝑤), 𝑧⟩)
2220, 21syl6eqr 2822 . . . . . . . . . . 11 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝐸)‘𝑣) = (((1st𝑌)‘𝑤)(1st𝐸)𝑧))
23 fveq2 6332 . . . . . . . . . . . 12 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝑍)‘𝑣) = ((1st𝑍)‘⟨((1st𝑌)‘𝑤), 𝑧⟩))
24 df-ov 6795 . . . . . . . . . . . 12 (((1st𝑌)‘𝑤)(1st𝑍)𝑧) = ((1st𝑍)‘⟨((1st𝑌)‘𝑤), 𝑧⟩)
2523, 24syl6eqr 2822 . . . . . . . . . . 11 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((1st𝑍)‘𝑣) = (((1st𝑌)‘𝑤)(1st𝑍)𝑧))
2622, 25oveq12d 6810 . . . . . . . . . 10 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)) = ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧)))
2719, 26eleq12d 2843 . . . . . . . . 9 (𝑣 = ⟨((1st𝑌)‘𝑤), 𝑧⟩ → ((𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)) ↔ (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧))))
28 yoneda.r . . . . . . . . . . . . . 14 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
2928fucbas 16826 . . . . . . . . . . . . 13 ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
30 yonedainv.i . . . . . . . . . . . . 13 𝐼 = (Inv‘𝑅)
31 yoneda.b . . . . . . . . . . . . . . . . . 18 𝐵 = (Base‘𝐶)
32 yoneda.1 . . . . . . . . . . . . . . . . . 18 1 = (Id‘𝐶)
33 yoneda.t . . . . . . . . . . . . . . . . . 18 𝑇 = (SetCat‘𝑉)
34 yoneda.h . . . . . . . . . . . . . . . . . 18 𝐻 = (HomF𝑄)
35 yoneda.e . . . . . . . . . . . . . . . . . 18 𝐸 = (𝑂 evalF 𝑆)
36 yoneda.z . . . . . . . . . . . . . . . . . 18 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
372, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8yonedalem1 17119 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
3837simpld 476 . . . . . . . . . . . . . . . 16 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
39 funcrcl 16729 . . . . . . . . . . . . . . . 16 (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
4038, 39syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
4140simpld 476 . . . . . . . . . . . . . 14 (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
4240simprd 477 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ Cat)
4328, 41, 42fuccat 16836 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ Cat)
4437simprd 477 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45 eqid 2770 . . . . . . . . . . . . 13 (Iso‘𝑅) = (Iso‘𝑅)
46 yoneda.m . . . . . . . . . . . . . 14 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
47 yonedainv.n . . . . . . . . . . . . . 14 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
482, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8, 46, 30, 47yonedainv 17128 . . . . . . . . . . . . 13 (𝜑𝑀(𝑍𝐼𝐸)𝑁)
4929, 30, 43, 38, 44, 45, 48inviso2 16633 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (𝐸(Iso‘𝑅)𝑍))
50 eqid 2770 . . . . . . . . . . . . . 14 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
516fucbas 16826 . . . . . . . . . . . . . 14 (𝑂 Func 𝑆) = (Base‘𝑄)
524, 31oppcbas 16584 . . . . . . . . . . . . . 14 𝐵 = (Base‘𝑂)
5350, 51, 52xpcbas 17025 . . . . . . . . . . . . 13 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
54 eqid 2770 . . . . . . . . . . . . 13 ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
55 eqid 2770 . . . . . . . . . . . . 13 (Iso‘𝑇) = (Iso‘𝑇)
5628, 53, 54, 44, 38, 45, 55fuciso 16841 . . . . . . . . . . . 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)))))
5749, 56mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣))))
5857simprd 477 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)))
5958adantr 466 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁𝑣) ∈ (((1st𝐸)‘𝑣)(Iso‘𝑇)((1st𝑍)‘𝑣)))
6031, 51, 16funcf1 16732 . . . . . . . . . . . 12 (𝜑 → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
6160adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
62 simprr 748 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑤𝐵)
6361, 62ffvelrnd 6503 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
64 simprl 746 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑧𝐵)
65 opelxpi 5288 . . . . . . . . . 10 ((((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆) ∧ 𝑧𝐵) → ⟨((1st𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
6663, 64, 65syl2anc 565 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ⟨((1st𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
6727, 59, 66rspcdva 3464 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st𝑌)‘𝑤)(1st𝐸)𝑧)(Iso‘𝑇)(((1st𝑌)‘𝑤)(1st𝑍)𝑧)))
684oppccat 16588 . . . . . . . . . . . . 13 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
693, 68syl 17 . . . . . . . . . . . 12 (𝜑𝑂 ∈ Cat)
7069adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑂 ∈ Cat)
715setccat 16941 . . . . . . . . . . . . 13 (𝑈 ∈ V → 𝑆 ∈ Cat)
7210, 71syl 17 . . . . . . . . . . . 12 (𝜑𝑆 ∈ Cat)
7372adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑆 ∈ Cat)
7435, 70, 73, 52, 63, 64evlf1 17067 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)(1st𝐸)𝑧) = ((1st ‘((1st𝑌)‘𝑤))‘𝑧))
753adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝐶 ∈ Cat)
76 eqid 2770 . . . . . . . . . . 11 (Hom ‘𝐶) = (Hom ‘𝐶)
772, 31, 75, 62, 76, 64yon11 17111 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
7874, 77eqtrd 2804 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)(1st𝐸)𝑧) = (𝑧(Hom ‘𝐶)𝑤))
797adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑉𝑊)
8011adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ran (Homf𝐶) ⊆ 𝑈)
818adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
822, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 75, 79, 80, 81, 63, 64yonedalem21 17120 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)(1st𝑍)𝑧) = (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
8378, 82oveq12d 6810 . . . . . . . 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))))
8467, 83eleqtrd 2851 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤))))
859adantr 466 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑈𝑉)
86 eqid 2770 . . . . . . . . . . . . 13 (Base‘𝑆) = (Base‘𝑆)
87 relfunc 16728 . . . . . . . . . . . . . 14 Rel (𝑂 Func 𝑆)
88 1st2ndbr 7365 . . . . . . . . . . . . . 14 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
8987, 63, 88sylancr 567 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st ‘((1st𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑤)))
9052, 86, 89funcf1 16732 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
9190, 64ffvelrnd 6503 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆))
925, 10setcbas 16934 . . . . . . . . . . . 12 (𝜑𝑈 = (Base‘𝑆))
9392adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑈 = (Base‘𝑆))
9491, 93eleqtrrd 2852 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
9577, 94eqeltrrd 2850 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈)
9685, 95sseldd 3751 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉)
97 eqid 2770 . . . . . . . . . 10 (Homf𝑄) = (Homf𝑄)
98 eqid 2770 . . . . . . . . . . 11 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
996, 98fuchom 16827 . . . . . . . . . 10 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
10061, 64ffvelrnd 6503 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
10197, 51, 99, 100, 63homfval 16558 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) = (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
1028unssad 3939 . . . . . . . . . . 11 (𝜑 → ran (Homf𝑄) ⊆ 𝑉)
103102adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ran (Homf𝑄) ⊆ 𝑉)
10497, 51homffn 16559 . . . . . . . . . . . 12 (Homf𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆))
105104a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (Homf𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)))
106 fnovrn 6955 . . . . . . . . . . 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))
107105, 100, 63, 106syl3anc 1475 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) ∈ ran (Homf𝑄))
108103, 107sseldd 3751 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(Homf𝑄)((1st𝑌)‘𝑤)) ∈ 𝑉)
109101, 108eqeltrrd 2850 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)) ∈ 𝑉)
11033, 79, 96, 109, 55setciso 16947 . . . . . . 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))))
11184, 110mpbid 222 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
11275adantr 466 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat)
113112adantr 466 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝐶 ∈ Cat)
11464adantr 466 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧𝐵)
115114adantr 466 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑧𝐵)
116 simpr 471 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑦𝐵)
1172, 31, 113, 115, 76, 116yon11 17111 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑧))‘𝑦) = (𝑦(Hom ‘𝐶)𝑧))
118117eqcomd 2776 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (𝑦(Hom ‘𝐶)𝑧) = ((1st ‘((1st𝑌)‘𝑧))‘𝑦))
119113adantr 466 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝐶 ∈ Cat)
12062ad3antrrr 701 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑤𝐵)
121115adantr 466 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧𝐵)
122 eqid 2770 . . . . . . . . . . . . . . 15 (comp‘𝐶) = (comp‘𝐶)
123116adantr 466 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦𝐵)
124 simpr 471 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
125 simpllr 752 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ∈ (𝑧(Hom ‘𝐶)𝑤))
1262, 31, 119, 120, 76, 121, 122, 123, 124, 125yon12 17112 . . . . . . . . . . . . . 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))
1272, 31, 119, 121, 76, 120, 122, 123, 125, 124yon2 17113 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔) = ((⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
128126, 127eqtr4d 2807 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘) = ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔))
129118, 128mpteq12dva 4864 . . . . . . . . . . . 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)))
13016adantr 466 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (1st𝑌)(𝐶 Func 𝑄)(2nd𝑌))
13131, 76, 99, 130, 64, 62funcf2 16734 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
132131ffvelrnda 6502 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) ∈ (((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
13398, 132nat1st2nd 16817 . . . . . . . . . . . . . . . 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))⟩))
134133adantr 466 . . . . . . . . . . . . . . 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))⟩))
135 eqid 2770 . . . . . . . . . . . . . . 15 (Hom ‘𝑆) = (Hom ‘𝑆)
13698, 134, 52, 135, 116natcl 16819 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (((𝑧(2nd𝑌)𝑤)‘)‘𝑦) ∈ (((1st ‘((1st𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑤))‘𝑦)))
13710adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → 𝑈 ∈ V)
138137ad2antrr 697 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑈 ∈ V)
13960ad2antrr 697 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st𝑌):𝐵⟶(𝑂 Func 𝑆))
140139, 114ffvelrnd 6503 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
141 1st2ndbr 7365 . . . . . . . . . . . . . . . . . . 19 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑧)))
14287, 140, 141sylancr 567 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑧)))
14352, 86, 142funcf1 16732 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑧)):𝐵⟶(Base‘𝑆))
144143ffvelrnda 6502 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆))
14593ad2antrr 697 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → 𝑈 = (Base‘𝑆))
146144, 145eleqtrrd 2852 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ∈ 𝑈)
14790adantr 466 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
148147ffvelrnda 6502 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆))
149148, 145eleqtrrd 2852 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → ((1st ‘((1st𝑌)‘𝑤))‘𝑦) ∈ 𝑈)
1505, 138, 135, 146, 149elsetchom 16937 . . . . . . . . . . . . . 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)))
151136, 150mpbid 222 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (((𝑧(2nd𝑌)𝑤)‘)‘𝑦):((1st ‘((1st𝑌)‘𝑧))‘𝑦)⟶((1st ‘((1st𝑌)‘𝑤))‘𝑦))
152151feqmptd 6391 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (((𝑧(2nd𝑌)𝑤)‘)‘𝑦) = (𝑔 ∈ ((1st ‘((1st𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd𝑌)𝑤)‘)‘𝑦)‘𝑔)))
153129, 152eqtr4d 2807 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘)) = (((𝑧(2nd𝑌)𝑤)‘)‘𝑦))
154153mpteq2dva 4876 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘))) = (𝑦𝐵 ↦ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦)))
15579adantr 466 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉𝑊)
15680adantr 466 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf𝐶) ⊆ 𝑈)
15781adantr 466 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
15863adantr 466 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
15977eleq2d 2835 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ( ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧) ↔ ∈ (𝑧(Hom ‘𝐶)𝑤)))
160159biimpar 463 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ∈ ((1st ‘((1st𝑌)‘𝑤))‘𝑧))
1612, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 112, 155, 156, 157, 158, 114, 47, 160yonedalem4a 17122 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st𝑌)‘𝑤)𝑁𝑧)‘) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st𝑌)‘𝑤))𝑦)‘𝑔)‘))))
16298, 133, 52natfn 16820 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) Fn 𝐵)
163 dffn5 6383 . . . . . . . . . . 11 (((𝑧(2nd𝑌)𝑤)‘) Fn 𝐵 ↔ ((𝑧(2nd𝑌)𝑤)‘) = (𝑦𝐵 ↦ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦)))
164162, 163sylib 208 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd𝑌)𝑤)‘) = (𝑦𝐵 ↦ (((𝑧(2nd𝑌)𝑤)‘)‘𝑦)))
165154, 161, 1643eqtr4d 2814 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵)) ∧ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st𝑌)‘𝑤)𝑁𝑧)‘) = ((𝑧(2nd𝑌)𝑤)‘))
166165mpteq2dva 4876 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st𝑌)‘𝑤)𝑁𝑧)‘)) = ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd𝑌)𝑤)‘)))
167 f1of 6278 . . . . . . . . . 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)))
168111, 167syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
169168feqmptd 6391 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) = ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st𝑌)‘𝑤)𝑁𝑧)‘)))
170131feqmptd 6391 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(2nd𝑌)𝑤) = ( ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd𝑌)𝑤)‘)))
171166, 169, 1703eqtr4d 2814 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (((1st𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd𝑌)𝑤))
172 f1oeq1 6268 . . . . . . 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))))
173171, 172syl 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))))
174111, 173mpbid 222 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
175174ralrimivva 3119 . . . 4 (𝜑 → ∀𝑧𝐵𝑤𝐵 (𝑧(2nd𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st𝑌)‘𝑤)))
17631, 76, 99isffth2 16782 . . . 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))))
17716, 175, 176sylanbrc 564 . . 3 (𝜑 → (1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌))
178 df-br 4785 . . 3 ((1st𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd𝑌) ↔ ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
179177, 178sylib 208 . 2 (𝜑 → ⟨(1st𝑌), (2nd𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
18014, 179eqeltrd 2849 1 (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144  ∀wral 3060  Vcvv 3349   ∪ cun 3719   ∩ cin 3720   ⊆ wss 3721  ⟨cop 4320   class class class wbr 4784   ↦ cmpt 4861   × cxp 5247  ran crn 5250  Rel wrel 5254   Fn wfn 6026  ⟶wf 6027  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794  1st c1st 7312  2nd c2nd 7313  tpos ctpos 7502  Basecbs 16063  Hom chom 16159  compcco 16160  Catccat 16531  Idccid 16532  Homf chomf 16533  oppCatcoppc 16577  Invcinv 16611  Isociso 16612   Func cfunc 16720   ∘func ccofu 16722   Full cful 16768   Faith cfth 16769   Nat cnat 16807   FuncCat cfuc 16808  SetCatcsetc 16931   ×c cxpc 17015   1stF c1stf 17016   2ndF c2ndf 17017   ⟨,⟩F cprf 17018   evalF cevlf 17056  HomFchof 17095  Yoncyon 17096 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-tpos 7503  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-hom 16173  df-cco 16174  df-cat 16535  df-cid 16536  df-homf 16537  df-comf 16538  df-oppc 16578  df-sect 16613  df-inv 16614  df-iso 16615  df-ssc 16676  df-resc 16677  df-subc 16678  df-func 16724  df-cofu 16726  df-full 16770  df-fth 16771  df-nat 16809  df-fuc 16810  df-setc 16932  df-xpc 17019  df-1stf 17020  df-2ndf 17021  df-prf 17022  df-evlf 17060  df-curf 17061  df-hof 17097  df-yon 17098 This theorem is referenced by:  yonffth  17131
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