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Theorem comfeq 16837
 Description: Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfeq.1 · = (comp‘𝐶)
comfeq.2 = (comp‘𝐷)
comfeq.h 𝐻 = (Hom ‘𝐶)
comfeq.3 (𝜑𝐵 = (Base‘𝐶))
comfeq.4 (𝜑𝐵 = (Base‘𝐷))
comfeq.5 (𝜑 → (Homf𝐶) = (Homf𝐷))
Assertion
Ref Expression
comfeq (𝜑 → ((compf𝐶) = (compf𝐷) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝑧,𝐵   𝐶,𝑓,𝑔,𝑧   𝜑,𝑓,𝑔,𝑧   · ,𝑓,𝑔,𝑥,𝑦   𝐷,𝑓,𝑔,𝑧   𝑓,𝐻,𝑔,𝑥,𝑦   ,𝑓,𝑔,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   (𝑧)   · (𝑧)   𝐻(𝑧)

Proof of Theorem comfeq
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ovex 7010 . . . . . 6 ((2nd𝑢)𝐻𝑧) ∈ V
2 fvex 6514 . . . . . 6 (𝐻𝑢) ∈ V
31, 2mpoex 7587 . . . . 5 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
43rgen2w 3101 . . . 4 𝑢 ∈ (𝐵 × 𝐵)∀𝑧𝐵 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
5 mpo2eqb 7101 . . . 4 (∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧𝐵 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V → ((𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) ↔ ∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧𝐵 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))))
64, 5ax-mp 5 . . 3 ((𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) ↔ ∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧𝐵 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)))
7 vex 3418 . . . . . . . . 9 𝑥 ∈ V
8 vex 3418 . . . . . . . . 9 𝑦 ∈ V
97, 8op2ndd 7514 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (2nd𝑢) = 𝑦)
109oveq1d 6993 . . . . . . 7 (𝑢 = ⟨𝑥, 𝑦⟩ → ((2nd𝑢)𝐻𝑧) = (𝑦𝐻𝑧))
11 fveq2 6501 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻𝑢) = (𝐻‘⟨𝑥, 𝑦⟩))
12 df-ov 6981 . . . . . . . . 9 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
1311, 12syl6eqr 2832 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻𝑢) = (𝑥𝐻𝑦))
14 oveq1 6985 . . . . . . . . . 10 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 · 𝑧) = (⟨𝑥, 𝑦· 𝑧))
1514oveqd 6995 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))
16 oveq1 6985 . . . . . . . . . 10 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 𝑧) = (⟨𝑥, 𝑦 𝑧))
1716oveqd 6995 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓))
1815, 17eqeq12d 2793 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓) ↔ (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
1913, 18raleqbidv 3341 . . . . . . 7 (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑓 ∈ (𝐻𝑢)(𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓) ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
2010, 19raleqbidv 3341 . . . . . 6 (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑔 ∈ ((2nd𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻𝑢)(𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓) ↔ ∀𝑔 ∈ (𝑦𝐻𝑧)∀𝑓 ∈ (𝑥𝐻𝑦)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
21 ovex 7010 . . . . . . . 8 (𝑔(𝑢 · 𝑧)𝑓) ∈ V
2221rgen2w 3101 . . . . . . 7 𝑔 ∈ ((2nd𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻𝑢)(𝑔(𝑢 · 𝑧)𝑓) ∈ V
23 mpo2eqb 7101 . . . . . . 7 (∀𝑔 ∈ ((2nd𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻𝑢)(𝑔(𝑢 · 𝑧)𝑓) ∈ V → ((𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)) ↔ ∀𝑔 ∈ ((2nd𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻𝑢)(𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓)))
2422, 23ax-mp 5 . . . . . 6 ((𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)) ↔ ∀𝑔 ∈ ((2nd𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻𝑢)(𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓))
25 ralcom 3295 . . . . . 6 (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓) ↔ ∀𝑔 ∈ (𝑦𝐻𝑧)∀𝑓 ∈ (𝑥𝐻𝑦)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓))
2620, 24, 253bitr4g 306 . . . . 5 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)) ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
2726ralbidv 3147 . . . 4 (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑧𝐵 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)) ↔ ∀𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
2827ralxp 5563 . . 3 (∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧𝐵 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓))
296, 28bitri 267 . 2 ((𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓))
30 comfeq.3 . . . . . 6 (𝜑𝐵 = (Base‘𝐶))
3130sqxpeqd 5440 . . . . 5 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
32 eqidd 2779 . . . . 5 (𝜑 → (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)))
3331, 30, 32mpoeq123dv 7049 . . . 4 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))))
34 eqid 2778 . . . . 5 (compf𝐶) = (compf𝐶)
35 eqid 2778 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
36 comfeq.h . . . . 5 𝐻 = (Hom ‘𝐶)
37 comfeq.1 . . . . 5 · = (comp‘𝐶)
3834, 35, 36, 37comfffval 16829 . . . 4 (compf𝐶) = (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)))
3933, 38syl6eqr 2832 . . 3 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (compf𝐶))
40 eqid 2778 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
41 comfeq.5 . . . . . . . . 9 (𝜑 → (Homf𝐶) = (Homf𝐷))
42413ad2ant1 1113 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (Homf𝐶) = (Homf𝐷))
43 xp2nd 7536 . . . . . . . . . 10 (𝑢 ∈ (𝐵 × 𝐵) → (2nd𝑢) ∈ 𝐵)
44433ad2ant2 1114 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (2nd𝑢) ∈ 𝐵)
45303ad2ant1 1113 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝐵 = (Base‘𝐶))
4644, 45eleqtrd 2868 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (2nd𝑢) ∈ (Base‘𝐶))
47 simp3 1118 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑧𝐵)
4847, 45eleqtrd 2868 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑧 ∈ (Base‘𝐶))
4935, 36, 40, 42, 46, 48homfeqval 16828 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((2nd𝑢)𝐻𝑧) = ((2nd𝑢)(Hom ‘𝐷)𝑧))
50 xp1st 7535 . . . . . . . . . . . 12 (𝑢 ∈ (𝐵 × 𝐵) → (1st𝑢) ∈ 𝐵)
51503ad2ant2 1114 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (1st𝑢) ∈ 𝐵)
5251, 45eleqtrd 2868 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (1st𝑢) ∈ (Base‘𝐶))
5335, 36, 40, 42, 52, 46homfeqval 16828 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((1st𝑢)𝐻(2nd𝑢)) = ((1st𝑢)(Hom ‘𝐷)(2nd𝑢)))
54 df-ov 6981 . . . . . . . . 9 ((1st𝑢)𝐻(2nd𝑢)) = (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩)
55 df-ov 6981 . . . . . . . . 9 ((1st𝑢)(Hom ‘𝐷)(2nd𝑢)) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩)
5653, 54, 553eqtr3g 2837 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩))
57 1st2nd2 7542 . . . . . . . . . 10 (𝑢 ∈ (𝐵 × 𝐵) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
58573ad2ant2 1114 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
5958fveq2d 6505 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻𝑢) = (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩))
6058fveq2d 6505 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((Hom ‘𝐷)‘𝑢) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩))
6156, 59, 603eqtr4d 2824 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻𝑢) = ((Hom ‘𝐷)‘𝑢))
62 eqidd 2779 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝑔(𝑢 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓))
6349, 61, 62mpoeq123dv 7049 . . . . . 6 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)))
6463mpoeq3dva 7051 . . . . 5 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))))
65 comfeq.4 . . . . . . 7 (𝜑𝐵 = (Base‘𝐷))
6665sqxpeqd 5440 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐷) × (Base‘𝐷)))
67 eqidd 2779 . . . . . 6 (𝜑 → (𝑔 ∈ ((2nd𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)))
6866, 65, 67mpoeq123dv 7049 . . . . 5 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ ((2nd𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))))
6964, 68eqtrd 2814 . . . 4 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ ((2nd𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))))
70 eqid 2778 . . . . 5 (compf𝐷) = (compf𝐷)
71 eqid 2778 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
72 comfeq.2 . . . . 5 = (comp‘𝐷)
7370, 71, 40, 72comfffval 16829 . . . 4 (compf𝐷) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ ((2nd𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)))
7469, 73syl6eqr 2832 . . 3 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) = (compf𝐷))
7539, 74eqeq12d 2793 . 2 (𝜑 → ((𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) ↔ (compf𝐶) = (compf𝐷)))
7629, 75syl5rbbr 278 1 (𝜑 → ((compf𝐶) = (compf𝐷) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  ∀wral 3088  Vcvv 3415  ⟨cop 4448   × cxp 5406  ‘cfv 6190  (class class class)co 6978   ∈ cmpo 6980  1st c1st 7501  2nd c2nd 7502  Basecbs 16342  Hom chom 16435  compcco 16436  Homf chomf 16798  compfccomf 16799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-1st 7503  df-2nd 7504  df-homf 16802  df-comf 16803 This theorem is referenced by:  comfeqd  16838  2oppccomf  16856  oppccomfpropd  16858  resssetc  17213  resscatc  17226
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