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Theorem yonedalem3 18169
Description: Lemma for yoneda 18172. (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 7390 . . . . . 6 (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ∈ V
32mptex 7173 . . . . 5 (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))) ∈ V
41, 3fnmpoi 8002 . . . 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 481 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝐶 ∈ Cat)
19 yoneda.w . . . . . . . . 9 (𝜑𝑉𝑊)
2019adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑉𝑊)
21 yoneda.u . . . . . . . . 9 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
2221adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → ran (Homf𝐶) ⊆ 𝑈)
23 yoneda.v . . . . . . . . 9 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
2423adantr 481 . . . . . . . 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 18163 . . . . . . 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 496 . . . . . 6 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔𝑀𝑦):(𝑔(1st𝑍)𝑦)⟶(𝑔(1st𝐸)𝑦))
29 eqid 2736 . . . . . . 7 (Hom ‘𝑇) = (Hom ‘𝑇)
30 eqid 2736 . . . . . . . . . . 11 (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
3112fucbas 17848 . . . . . . . . . . 11 (𝑂 Func 𝑆) = (Base‘𝑄)
329, 7oppcbas 17599 . . . . . . . . . . 11 𝐵 = (Base‘𝑂)
3330, 31, 32xpcbas 18066 . . . . . . . . . 10 ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
34 eqid 2736 . . . . . . . . . 10 (Base‘𝑇) = (Base‘𝑇)
35 relfunc 17748 . . . . . . . . . . 11 Rel ((𝑄 ×c 𝑂) Func 𝑇)
366, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23yonedalem1 18161 . . . . . . . . . . . 12 (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
3736simpld 495 . . . . . . . . . . 11 (𝜑𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
38 1st2ndbr 7974 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
3935, 37, 38sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝑍))
4033, 34, 39funcf1 17752 . . . . . . . . 9 (𝜑 → (1st𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
4140fovcdmda 7525 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝑍)𝑦) ∈ (Base‘𝑇))
4211, 20setcbas 17964 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → 𝑉 = (Base‘𝑇))
4341, 42eleqtrrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝑍)𝑦) ∈ 𝑉)
4436simprd 496 . . . . . . . . . . 11 (𝜑𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45 1st2ndbr 7974 . . . . . . . . . . 11 ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
4635, 44, 45sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd𝐸))
4733, 34, 46funcf1 17752 . . . . . . . . 9 (𝜑 → (1st𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
4847fovcdmda 7525 . . . . . . . 8 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝐸)𝑦) ∈ (Base‘𝑇))
4948, 42eleqtrrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔(1st𝐸)𝑦) ∈ 𝑉)
5011, 20, 29, 43, 49elsetchom 17967 . . . . . 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 256 . . . . 5 ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦𝐵)) → (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
5251ralrimivva 3197 . . . 4 (𝜑 → ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
53 fveq2 6842 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀𝑧) = (𝑀‘⟨𝑔, 𝑦⟩))
54 df-ov 7360 . . . . . . 7 (𝑔𝑀𝑦) = (𝑀‘⟨𝑔, 𝑦⟩)
5553, 54eqtr4di 2794 . . . . . 6 (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀𝑧) = (𝑔𝑀𝑦))
56 fveq2 6842 . . . . . . . 8 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝑍)‘𝑧) = ((1st𝑍)‘⟨𝑔, 𝑦⟩))
57 df-ov 7360 . . . . . . . 8 (𝑔(1st𝑍)𝑦) = ((1st𝑍)‘⟨𝑔, 𝑦⟩)
5856, 57eqtr4di 2794 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝑍)‘𝑧) = (𝑔(1st𝑍)𝑦))
59 fveq2 6842 . . . . . . . 8 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝐸)‘𝑧) = ((1st𝐸)‘⟨𝑔, 𝑦⟩))
60 df-ov 7360 . . . . . . . 8 (𝑔(1st𝐸)𝑦) = ((1st𝐸)‘⟨𝑔, 𝑦⟩)
6159, 60eqtr4di 2794 . . . . . . 7 (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st𝐸)‘𝑧) = (𝑔(1st𝐸)𝑦))
6258, 61oveq12d 7375 . . . . . 6 (𝑧 = ⟨𝑔, 𝑦⟩ → (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) = ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦)))
6355, 62eleq12d 2832 . . . . 5 (𝑧 = ⟨𝑔, 𝑦⟩ → ((𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)) ↔ (𝑔𝑀𝑦) ∈ ((𝑔(1st𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st𝐸)𝑦))))
6463ralxp 5797 . . . 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 233 . . 3 (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀𝑧) ∈ (((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)))
66 ovex 7390 . . . . . 6 (𝑂 Func 𝑆) ∈ V
677fvexi 6856 . . . . . 6 𝐵 ∈ V
6866, 67mpoex 8012 . . . . 5 (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥)))) ∈ V
691, 68eqeltri 2834 . . . 4 𝑀 ∈ V
7069elixp 8842 . . 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 583 . 2 (𝜑𝑀X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st𝑍)‘𝑧)(Hom ‘𝑇)((1st𝐸)‘𝑧)))
7217adantr 481 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝐶 ∈ Cat)
7319adantr 481 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑉𝑊)
7421adantr 481 . . . . 5 ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ran (Homf𝐶) ⊆ 𝑈)
7523adantr 481 . . . . 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 1194 . . . . . 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 7953 . . . . . 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 7954 . . . . . 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 1195 . . . . . 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 7953 . . . . . 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 7954 . . . . . 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 1196 . . . . . . 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 2736 . . . . . . . . . 10 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
8812, 87fuchom 17849 . . . . . . . . 9 (𝑂 Nat 𝑆) = (Hom ‘𝑄)
89 eqid 2736 . . . . . . . . 9 (Hom ‘𝑂) = (Hom ‘𝑂)
90 eqid 2736 . . . . . . . . 9 (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
9130, 33, 88, 89, 90, 76, 81xpchom 18068 . . . . . . . 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 2736 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
9392, 9oppchom 17596 . . . . . . . . 9 ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤)) = ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧))
9493xpeq2i 5660 . . . . . . . 8 (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑧)(Hom ‘𝑂)(2nd𝑤))) = (((1st𝑧)(𝑂 Nat 𝑆)(1st𝑤)) × ((2nd𝑤)(Hom ‘𝐶)(2nd𝑧)))
9591, 94eqtrdi 2792 . . . . . . 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 2840 . . . . . 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 7953 . . . . . 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 7954 . . . . . 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 18168 . . . 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 7960 . . . . . . . . . 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 6846 . . . . . . . 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 7360 . . . . . . . 8 ((1st𝑧)(1st𝑍)(2nd𝑧)) = ((1st𝑍)‘⟨(1st𝑧), (2nd𝑧)⟩)
106104, 105eqtr4di 2794 . . . . . . 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 7960 . . . . . . . . . 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 6846 . . . . . . . 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 7360 . . . . . . . 8 ((1st𝑤)(1st𝑍)(2nd𝑤)) = ((1st𝑍)‘⟨(1st𝑤), (2nd𝑤)⟩)
111109, 110eqtr4di 2794 . . . . . . 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 4838 . . . . . 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 6846 . . . . . . 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 7360 . . . . . . 7 ((1st𝑤)(1st𝐸)(2nd𝑤)) = ((1st𝐸)‘⟨(1st𝑤), (2nd𝑤)⟩)
115113, 114eqtr4di 2794 . . . . . 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 7375 . . . . 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 6846 . . . . . 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 7360 . . . . . 6 ((1st𝑤)𝑀(2nd𝑤)) = (𝑀‘⟨(1st𝑤), (2nd𝑤)⟩)
119117, 118eqtr4di 2794 . . . . 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 7375 . . . . . . 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 7960 . . . . . . . 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 6849 . . . . . 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 7360 . . . . . 6 ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝑍)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
125123, 124eqtr4di 2794 . . . . 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 7378 . . . 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 6846 . . . . . . . 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 7360 . . . . . . . 8 ((1st𝑧)(1st𝐸)(2nd𝑧)) = ((1st𝐸)‘⟨(1st𝑧), (2nd𝑧)⟩)
129127, 128eqtr4di 2794 . . . . . . 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 4838 . . . . . 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 7375 . . . . 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 7375 . . . . . . 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 6849 . . . . . 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 7360 . . . . . 6 ((1st𝑔)(⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)(2nd𝑔)) = ((⟨(1st𝑧), (2nd𝑧)⟩(2nd𝐸)⟨(1st𝑤), (2nd𝑤)⟩)‘⟨(1st𝑔), (2nd𝑔)⟩)
135133, 134eqtr4di 2794 . . . . 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 6846 . . . . . 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 7360 . . . . . 6 ((1st𝑧)𝑀(2nd𝑧)) = (𝑀‘⟨(1st𝑧), (2nd𝑧)⟩)
138136, 137eqtr4di 2794 . . . . 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 7378 . . . 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 2786 . . 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 3200 . 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 2736 . . 3 ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
143 eqid 2736 . . 3 (comp‘𝑇) = (comp‘𝑇)
144142, 33, 90, 29, 143, 37, 44isnat2 17835 . 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cun 3908  wss 3910  cop 4592   class class class wbr 5105  cmpt 5188   × cxp 5631  ran crn 5634  Rel wrel 5638   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  tpos ctpos 8156  Xcixp 8835  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Idccid 17545  Homf chomf 17546  oppCatcoppc 17591   Func cfunc 17740  func ccofu 17742   Nat cnat 17828   FuncCat cfuc 17829  SetCatcsetc 17961   ×c cxpc 18056   1stF c1stf 18057   2ndF c2ndf 18058   ⟨,⟩F cprf 18059   evalF cevlf 18098  HomFchof 18137  Yoncyon 18138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-homf 17550  df-comf 17551  df-oppc 17592  df-ssc 17693  df-resc 17694  df-subc 17695  df-func 17744  df-cofu 17746  df-nat 17830  df-fuc 17831  df-setc 17962  df-xpc 18060  df-1stf 18061  df-2ndf 18062  df-prf 18063  df-evlf 18102  df-curf 18103  df-hof 18139  df-yon 18140
This theorem is referenced by:  yonedainv  18170
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