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Theorem yonedalem3 17524
Description: Lemma for yoneda 17527. (Contributed by Mario Carneiro, 28-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))))
Assertion
Ref Expression
yonedalem3 (𝜑𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))
Distinct variable groups:   𝑓,𝑎,𝑥, 1   𝐶,𝑎,𝑓,𝑥   𝐸,𝑎,𝑓   𝐵,𝑎,𝑓,𝑥   𝑂,𝑎,𝑓,𝑥   𝑆,𝑎,𝑓,𝑥   𝑄,𝑎,𝑓,𝑥   𝑇,𝑓   𝜑,𝑎,𝑓,𝑥   𝑌,𝑎,𝑓,𝑥   𝑍,𝑎,𝑓,𝑥
Allowed substitution hints:   𝑅(𝑥,𝑓,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑓,𝑎)   𝐸(𝑥)   𝐻(𝑥,𝑓,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑉(𝑥,𝑓,𝑎)   𝑊(𝑥,𝑓,𝑎)

Proof of Theorem yonedalem3
Dummy variables 𝑔 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.m . . . . 5 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))
2 ovex 7183 . . . . . 6 (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ∈ V
32mptex 6980 . . . . 5 (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))) ∈ V
41, 3fnmpoi 7762 . . . 4 𝑀 Fn ((𝑂 Func 𝑆) × 𝐵)
54a1i 11 . . 3 (𝜑𝑀 Fn ((𝑂 Func 𝑆) × 𝐵))
6 yoneda.y . . . . . . . 8 𝑌 = (Yon‘𝐶)
7 yoneda.b . . . . . . . 8 𝐵 = (Base‘𝐶)
8 yoneda.1 . . . . . . . 8 1 = (Id‘𝐶)
9 yoneda.o . . . . . . . 8 𝑂 = (oppCat‘𝐶)
10 yoneda.s . . . . . . . 8 𝑆 = (SetCat‘𝑈)
11 yoneda.t . . . . . . . 8 𝑇 = (SetCat‘𝑉)
12 yoneda.q . . . . . . . 8 𝑄 = (𝑂 FuncCat 𝑆)
13 yoneda.h . . . . . . . 8 𝐻 = (HomF𝑄)
14 yoneda.r . . . . . . . 8 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
15 yoneda.e . . . . . . . 8 𝐸 = (𝑂 evalF 𝑆)
16 yoneda.z . . . . . . . 8 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17 yoneda.c . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
1817adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝐶 ∈ Cat)
19 yoneda.w . . . . . . . . 9 (𝜑𝑉𝑊)
2019adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑉𝑊)
21 yoneda.u . . . . . . . . 9 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2221adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → ran (Homf𝐶) ⊆ 𝑈)
23 yoneda.v . . . . . . . . 9 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2423adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
25 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑔 ∈ (𝑂 Func 𝑆))
26 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑦𝐵)
276, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 1yonedalem3a 17518 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → ((𝑔𝑀𝑦) = (𝑎 ∈ (((1st𝑌)‘𝑦)(𝑂 Nat 𝑆)𝑔) ↦ ((𝑎𝑦)‘( 1𝑦))) ∧ (𝑔𝑀𝑦):(𝑔(1st𝑍)𝑦)⟶(𝑔(1st𝐸)𝑦)))
2827simprd 498 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔𝑀𝑦):(𝑔(1st𝑍)𝑦)⟶(𝑔(1st𝐸)𝑦))
29 eqid 2821 . . . . . . 7 (Hom ‘𝑇) = (Hom ‘𝑇)
30 eqid 2821 . . . . . . . . . . 11 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
3112fucbas 17224 . . . . . . . . . . 11 (𝑂 Func 𝑆) = (Base‘𝑄)
329, 7oppcbas 16982 . . . . . . . . . . 11 𝐵 = (Base‘𝑂)
3330, 31, 32xpcbas 17422 . . . . . . . . . 10 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
34 eqid 2821 . . . . . . . . . 10 (Base‘𝑇) = (Base‘𝑇)
35 relfunc 17126 . . . . . . . . . . 11 Rel ((𝑄 ×c 𝑂) Func 𝑇)
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 17516 . . . . . . . . . . . 12 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
3736simpld 497 . . . . . . . . . . 11 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
38 1st2ndbr 7735 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
3935, 37, 38sylancr 589 . . . . . . . . . 10 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
4033, 34, 39funcf1 17130 . . . . . . . . 9 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
4140fovrnda 7313 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝑍)𝑦) ∈ (Base‘𝑇))
4211, 20setcbas 17332 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑉 = (Base‘𝑇))
4341, 42eleqtrrd 2916 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝑍)𝑦) ∈ 𝑉)
4436simprd 498 . . . . . . . . . . 11 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45 1st2ndbr 7735 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
4635, 44, 45sylancr 589 . . . . . . . . . 10 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
4733, 34, 46funcf1 17130 . . . . . . . . 9 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
4847fovrnda 7313 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝐸)𝑦) ∈ (Base‘𝑇))
4948, 42eleqtrrd 2916 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝐸)𝑦) ∈ 𝑉)
5011, 20, 29, 43, 49elsetchom 17335 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → ((𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)) ↔ (𝑔𝑀𝑦):(𝑔(1st𝑍)𝑦)⟶(𝑔(1st𝐸)𝑦)))
5128, 50mpbird 259 . . . . 5 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
5251ralrimivva 3191 . . . 4 (𝜑 → ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
53 fveq2 6664 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀𝑧) = (𝑀‘⟨𝑔, 𝑦⟩))
54 df-ov 7153 . . . . . . 7 (𝑔𝑀𝑦) = (𝑀‘⟨𝑔, 𝑦⟩)
5553, 54syl6eqr 2874 . . . . . 6 (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀𝑧) = (𝑔𝑀𝑦))
56 fveq2 6664 . . . . . . . 8 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝑍)‘𝑧) = ((1st𝑍)‘⟨𝑔, 𝑦⟩))
57 df-ov 7153 . . . . . . . 8 (𝑔(1st𝑍)𝑦) = ((1st𝑍)‘⟨𝑔, 𝑦⟩)
5856, 57syl6eqr 2874 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝑍)‘𝑧) = (𝑔(1st𝑍)𝑦))
59 fveq2 6664 . . . . . . . 8 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨𝑔, 𝑦⟩))
60 df-ov 7153 . . . . . . . 8 (𝑔(1st𝐸)𝑦) = ((1st𝐸)‘⟨𝑔, 𝑦⟩)
6159, 60syl6eqr 2874 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝐸)‘𝑧) = (𝑔(1st𝐸)𝑦))
6258, 61oveq12d 7168 . . . . . 6 (𝑧 = ⟨𝑔, 𝑦⟩ → (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) = ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
6355, 62eleq12d 2907 . . . . 5 (𝑧 = ⟨𝑔, 𝑦⟩ → ((𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ↔ (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦))))
6463ralxp 5706 . . . 4 (∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ↔ ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
6552, 64sylibr 236 . . 3 (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)))
66 ovex 7183 . . . . . 6 (𝑂 Func 𝑆) ∈ V
677fvexi 6678 . . . . . 6 𝐵 ∈ V
6866, 67mpoex 7771 . . . . 5 (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥)))) ∈ V
691, 68eqeltri 2909 . . . 4 𝑀 ∈ V
7069elixp 8462 . . 3 (𝑀X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ↔ (𝑀 Fn ((𝑂 Func 𝑆) × 𝐵) ∧ ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧))))
715, 65, 70sylanbrc 585 . 2 (𝜑𝑀X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)))
7217adantr 483 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝐶 ∈ Cat)
7319adantr 483 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑉𝑊)
7421adantr 483 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ran (Homf𝐶) ⊆ 𝑈)
7523adantr 483 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
76 simpr1 1190 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵))
77 xp1st 7715 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1st𝑧) ∈ (𝑂 Func 𝑆))
7876, 77syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st𝑧) ∈ (𝑂 Func 𝑆))
79 xp2nd 7716 . . . . . 6 (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2nd𝑧) ∈ 𝐵)
8076, 79syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd𝑧) ∈ 𝐵)
81 simpr2 1191 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵))
82 xp1st 7715 . . . . . 6 (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1st𝑤) ∈ (𝑂 Func 𝑆))
8381, 82syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st𝑤) ∈ (𝑂 Func 𝑆))
84 xp2nd 7716 . . . . . 6 (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2nd𝑤) ∈ 𝐵)
8581, 84syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd𝑤) ∈ 𝐵)
86 simpr3 1192 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))
87 eqid 2821 . . . . . . . . . 10 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8812, 87fuchom 17225 . . . . . . . . 9 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
89 eqid 2821 . . . . . . . . 9 (Hom ‘𝑂) = (Hom ‘𝑂)
90 eqid 2821 . . . . . . . . 9 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
9130, 33, 88, 89, 90, 76, 81xpchom 17424 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤) = (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤))))
92 eqid 2821 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
9392, 9oppchom 16979 . . . . . . . . 9 ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤)) = ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))
9493xpeq2i 5576 . . . . . . . 8 (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤))) = (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧)))
9591, 94syl6eq 2872 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤) = (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))))
9686, 95eleqtrd 2915 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))))
97 xp1st 7715 . . . . . 6 (𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))) → (1st𝑔) ∈ ((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)))
9896, 97syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st𝑔) ∈ ((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)))
99 xp2nd 7716 . . . . . 6 (𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))) → (2nd𝑔) ∈ ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧)))
10096, 99syl 17 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd𝑔) ∈ ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧)))
1016, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 72, 73, 74, 75, 78, 80, 83, 85, 98, 100, 1yonedalem3b 17523 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (((1st𝑤)𝑀(2nd𝑤))(⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑤)(1st𝑍)(2nd𝑤))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤)))((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔))) = (((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔))(⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑧)(1st𝐸)(2nd𝑧))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤)))((1st𝑧)𝑀(2nd𝑧))))
102 1st2nd2 7722 . . . . . . . . . 10 (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
10376, 102syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
104103fveq2d 6668 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑧) = ((1st𝑍)‘⟨(1st𝑧), (2nd𝑧)⟩))
105 df-ov 7153 . . . . . . . 8 ((1st𝑧)(1st𝑍)(2nd𝑧)) = ((1st𝑍)‘⟨(1st𝑧), (2nd𝑧)⟩)
106104, 105syl6eqr 2874 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑧) = ((1st𝑧)(1st𝑍)(2nd𝑧)))
107 1st2nd2 7722 . . . . . . . . . 10 (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
10881, 107syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
109108fveq2d 6668 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑤) = ((1st𝑍)‘⟨(1st𝑤), (2nd𝑤)⟩))
110 df-ov 7153 . . . . . . . 8 ((1st𝑤)(1st𝑍)(2nd𝑤)) = ((1st𝑍)‘⟨(1st𝑤), (2nd𝑤)⟩)
111109, 110syl6eqr 2874 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝑍)‘𝑤) = ((1st𝑤)(1st𝑍)(2nd𝑤)))
112106, 111opeq12d 4804 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩ = ⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑤)(1st𝑍)(2nd𝑤))⟩)
113108fveq2d 6668 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑤) = ((1st𝐸)‘⟨(1st𝑤), (2nd𝑤)⟩))
114 df-ov 7153 . . . . . . 7 ((1st𝑤)(1st𝐸)(2nd𝑤)) = ((1st𝐸)‘⟨(1st𝑤), (2nd𝑤)⟩)
115113, 114syl6eqr 2874 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑤) = ((1st𝑤)(1st𝐸)(2nd𝑤)))
116112, 115oveq12d 7168 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤)) = (⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑤)(1st𝑍)(2nd𝑤))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤))))
117108fveq2d 6668 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑤) = (𝑀‘⟨(1st𝑤), (2nd𝑤)⟩))
118 df-ov 7153 . . . . . 6 ((1st𝑤)𝑀(2nd𝑤)) = (𝑀‘⟨(1st𝑤), (2nd𝑤)⟩)
119117, 118syl6eqr 2874 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑤) = ((1st𝑤)𝑀(2nd𝑤)))
120103, 108oveq12d 7168 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(2nd𝑍)𝑤) = (⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩))
121 1st2nd2 7722 . . . . . . . 8 (𝑔 ∈ (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
12296, 121syl 17 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 = ⟨(1st𝑔), (2nd𝑔)⟩)
123120, 122fveq12d 6671 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd𝑍)𝑤)‘𝑔) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩))
124 df-ov 7153 . . . . . 6 ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
125123, 124syl6eqr 2874 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd𝑍)𝑤)‘𝑔) = ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)))
126116, 119, 125oveq123d 7171 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑀𝑤)(⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤))((𝑧(2nd𝑍)𝑤)‘𝑔)) = (((1st𝑤)𝑀(2nd𝑤))(⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑤)(1st𝑍)(2nd𝑤))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤)))((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔))))
127103fveq2d 6668 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩))
128 df-ov 7153 . . . . . . . 8 ((1st𝑧)(1st𝐸)(2nd𝑧)) = ((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩)
129127, 128syl6eqr 2874 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st𝐸)‘𝑧) = ((1st𝑧)(1st𝐸)(2nd𝑧)))
130106, 129opeq12d 4804 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩ = ⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑧)(1st𝐸)(2nd𝑧))⟩)
131130, 115oveq12d 7168 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤)) = (⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑧)(1st𝐸)(2nd𝑧))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤))))
132103, 108oveq12d 7168 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(2nd𝐸)𝑤) = (⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩))
133132, 122fveq12d 6671 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd𝐸)𝑤)‘𝑔) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩))
134 df-ov 7153 . . . . . 6 ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
135133, 134syl6eqr 2874 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd𝐸)𝑤)‘𝑔) = ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)))
136103fveq2d 6668 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑧) = (𝑀‘⟨(1st𝑧), (2nd𝑧)⟩))
137 df-ov 7153 . . . . . 6 ((1st𝑧)𝑀(2nd𝑧)) = (𝑀‘⟨(1st𝑧), (2nd𝑧)⟩)
138136, 137syl6eqr 2874 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀𝑧) = ((1st𝑧)𝑀(2nd𝑧)))
139131, 135, 138oveq123d 7171 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (((𝑧(2nd𝐸)𝑤)‘𝑔)(⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤))(𝑀𝑧)) = (((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔))(⟨((1st𝑧)(1st𝑍)(2nd𝑧)), ((1st𝑧)(1st𝐸)(2nd𝑧))⟩(comp‘𝑇)((1st𝑤)(1st𝐸)(2nd𝑤)))((1st𝑧)𝑀(2nd𝑧))))
140101, 126, 1393eqtr4d 2866 . . 3 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑀𝑤)(⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤))((𝑧(2nd𝑍)𝑤)‘𝑔)) = (((𝑧(2nd𝐸)𝑤)‘𝑔)(⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤))(𝑀𝑧)))
141140ralrimivvva 3192 . 2 (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤)((𝑀𝑤)(⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤))((𝑧(2nd𝑍)𝑤)‘𝑔)) = (((𝑧(2nd𝐸)𝑤)‘𝑔)(⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤))(𝑀𝑧)))
142 eqid 2821 . . 3 ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
143 eqid 2821 . . 3 (comp‘𝑇) = (comp‘𝑇)
144142, 33, 90, 29, 143, 37, 44isnat2 17212 . 2 (𝜑 → (𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸) ↔ (𝑀X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ∧ ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤)((𝑀𝑤)(⟨((1st𝑍)‘𝑧), ((1st𝑍)‘𝑤)⟩(comp‘𝑇)((1st𝐸)‘𝑤))((𝑧(2nd𝑍)𝑤)‘𝑔)) = (((𝑧(2nd𝐸)𝑤)‘𝑔)(⟨((1st𝑍)‘𝑧), ((1st𝐸)‘𝑧)⟩(comp‘𝑇)((1st𝐸)‘𝑤))(𝑀𝑧)))))
14571, 141, 144mpbir2and 711 1 (𝜑𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  Vcvv 3494  cun 3933  wss 3935  cop 4566   class class class wbr 5058  cmpt 5138   × cxp 5547  ran crn 5550  Rel wrel 5554   Fn wfn 6344  wf 6345  cfv 6349  (class class class)co 7150  cmpo 7152  1st c1st 7681  2nd c2nd 7682  tpos ctpos 7885  Xcixp 8455  Basecbs 16477  Hom chom 16570  compcco 16571  Catccat 16929  Idccid 16930  Homf chomf 16931  oppCatcoppc 16975   Func cfunc 17118  func ccofu 17120   Nat cnat 17205   FuncCat cfuc 17206  SetCatcsetc 17329   ×c cxpc 17412   1stF c1stf 17413   2ndF c2ndf 17414   ⟨,⟩F cprf 17415   evalF cevlf 17453  HomFchof 17492  Yoncyon 17493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-hom 16583  df-cco 16584  df-cat 16933  df-cid 16934  df-homf 16935  df-comf 16936  df-oppc 16976  df-ssc 17074  df-resc 17075  df-subc 17076  df-func 17122  df-cofu 17124  df-nat 17207  df-fuc 17208  df-setc 17330  df-xpc 17416  df-1stf 17417  df-2ndf 17418  df-prf 17419  df-evlf 17457  df-curf 17458  df-hof 17494  df-yon 17495
This theorem is referenced by:  yonedainv  17525
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