MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  comfeq Structured version   Visualization version   GIF version

Theorem comfeq 16567
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 6841 . . . . . 6 ((2nd𝑢)𝐻𝑧) ∈ V
2 fvex 6362 . . . . . 6 (𝐻𝑢) ∈ V
31, 2mpt2ex 7415 . . . . 5 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
43rgen2w 3063 . . . 4 𝑢 ∈ (𝐵 × 𝐵)∀𝑧𝐵 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
5 mpt22eqb 6934 . . . 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 3343 . . . . . . . . 9 𝑥 ∈ V
8 vex 3343 . . . . . . . . 9 𝑦 ∈ V
97, 8op2ndd 7344 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (2nd𝑢) = 𝑦)
109oveq1d 6828 . . . . . . 7 (𝑢 = ⟨𝑥, 𝑦⟩ → ((2nd𝑢)𝐻𝑧) = (𝑦𝐻𝑧))
11 fveq2 6352 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻𝑢) = (𝐻‘⟨𝑥, 𝑦⟩))
12 df-ov 6816 . . . . . . . . 9 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
1311, 12syl6eqr 2812 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻𝑢) = (𝑥𝐻𝑦))
14 oveq1 6820 . . . . . . . . . 10 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 · 𝑧) = (⟨𝑥, 𝑦· 𝑧))
1514oveqd 6830 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))
16 oveq1 6820 . . . . . . . . . 10 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 𝑧) = (⟨𝑥, 𝑦 𝑧))
1716oveqd 6830 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓))
1815, 17eqeq12d 2775 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓) ↔ (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
1913, 18raleqbidv 3291 . . . . . . 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 3291 . . . . . 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 6841 . . . . . . . 8 (𝑔(𝑢 · 𝑧)𝑓) ∈ V
2221rgen2w 3063 . . . . . . 7 𝑔 ∈ ((2nd𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻𝑢)(𝑔(𝑢 · 𝑧)𝑓) ∈ V
23 mpt22eqb 6934 . . . . . . 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 3236 . . . . . 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 303 . . . . 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 3124 . . . 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 5419 . . 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 264 . 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 5298 . . . . 5 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
32 eqidd 2761 . . . . 5 (𝜑 → (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)))
3331, 30, 32mpt2eq123dv 6882 . . . 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 2760 . . . . 5 (compf𝐶) = (compf𝐶)
35 eqid 2760 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
36 comfeq.h . . . . 5 𝐻 = (Hom ‘𝐶)
37 comfeq.1 . . . . 5 · = (comp‘𝐶)
3834, 35, 36, 37comfffval 16559 . . . 4 (compf𝐶) = (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)))
3933, 38syl6eqr 2812 . . 3 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (compf𝐶))
40 eqid 2760 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
41 comfeq.5 . . . . . . . . 9 (𝜑 → (Homf𝐶) = (Homf𝐷))
42413ad2ant1 1128 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (Homf𝐶) = (Homf𝐷))
43 xp2nd 7366 . . . . . . . . . 10 (𝑢 ∈ (𝐵 × 𝐵) → (2nd𝑢) ∈ 𝐵)
44433ad2ant2 1129 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (2nd𝑢) ∈ 𝐵)
45303ad2ant1 1128 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝐵 = (Base‘𝐶))
4644, 45eleqtrd 2841 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (2nd𝑢) ∈ (Base‘𝐶))
47 simp3 1133 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑧𝐵)
4847, 45eleqtrd 2841 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑧 ∈ (Base‘𝐶))
4935, 36, 40, 42, 46, 48homfeqval 16558 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((2nd𝑢)𝐻𝑧) = ((2nd𝑢)(Hom ‘𝐷)𝑧))
50 xp1st 7365 . . . . . . . . . . . 12 (𝑢 ∈ (𝐵 × 𝐵) → (1st𝑢) ∈ 𝐵)
51503ad2ant2 1129 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (1st𝑢) ∈ 𝐵)
5251, 45eleqtrd 2841 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (1st𝑢) ∈ (Base‘𝐶))
5335, 36, 40, 42, 52, 46homfeqval 16558 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((1st𝑢)𝐻(2nd𝑢)) = ((1st𝑢)(Hom ‘𝐷)(2nd𝑢)))
54 df-ov 6816 . . . . . . . . 9 ((1st𝑢)𝐻(2nd𝑢)) = (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩)
55 df-ov 6816 . . . . . . . . 9 ((1st𝑢)(Hom ‘𝐷)(2nd𝑢)) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩)
5653, 54, 553eqtr3g 2817 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩))
57 1st2nd2 7372 . . . . . . . . . 10 (𝑢 ∈ (𝐵 × 𝐵) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
58573ad2ant2 1129 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
5958fveq2d 6356 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻𝑢) = (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩))
6058fveq2d 6356 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((Hom ‘𝐷)‘𝑢) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩))
6156, 59, 603eqtr4d 2804 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻𝑢) = ((Hom ‘𝐷)‘𝑢))
62 eqidd 2761 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝑔(𝑢 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓))
6349, 61, 62mpt2eq123dv 6882 . . . . . 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)))
6463mpt2eq3dva 6884 . . . . 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 5298 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐷) × (Base‘𝐷)))
67 eqidd 2761 . . . . . 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, 67mpt2eq123dv 6882 . . . . 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 2794 . . . 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 2760 . . . . 5 (compf𝐷) = (compf𝐷)
71 eqid 2760 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
72 comfeq.2 . . . . 5 = (comp‘𝐷)
7370, 71, 40, 72comfffval 16559 . . . 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 2812 . . 3 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) = (compf𝐷))
7539, 74eqeq12d 2775 . 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 275 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 196  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cop 4327   × cxp 5264  cfv 6049  (class class class)co 6813  cmpt2 6815  1st c1st 7331  2nd c2nd 7332  Basecbs 16059  Hom chom 16154  compcco 16155  Homf chomf 16528  compfccomf 16529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-homf 16532  df-comf 16533
This theorem is referenced by:  comfeqd  16568  2oppccomf  16586  oppccomfpropd  16588  resssetc  16943  resscatc  16956
  Copyright terms: Public domain W3C validator