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Theorem comfeq 17749
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 comfeq.3 . . . . . 6 (𝜑𝐵 = (Base‘𝐶))
21sqxpeqd 5717 . . . . 5 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
3 eqidd 2738 . . . . 5 (𝜑 → (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)))
42, 1, 3mpoeq123dv 7508 . . . 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))))
5 eqid 2737 . . . . 5 (compf𝐶) = (compf𝐶)
6 eqid 2737 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
7 comfeq.h . . . . 5 𝐻 = (Hom ‘𝐶)
8 comfeq.1 . . . . 5 · = (comp‘𝐶)
95, 6, 7, 8comfffval 17741 . . . 4 (compf𝐶) = (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)))
104, 9eqtr4di 2795 . . 3 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (compf𝐶))
11 eqid 2737 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
12 comfeq.5 . . . . . . . . 9 (𝜑 → (Homf𝐶) = (Homf𝐷))
13123ad2ant1 1134 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (Homf𝐶) = (Homf𝐷))
14 xp2nd 8047 . . . . . . . . . 10 (𝑢 ∈ (𝐵 × 𝐵) → (2nd𝑢) ∈ 𝐵)
15143ad2ant2 1135 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (2nd𝑢) ∈ 𝐵)
1613ad2ant1 1134 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝐵 = (Base‘𝐶))
1715, 16eleqtrd 2843 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (2nd𝑢) ∈ (Base‘𝐶))
18 simp3 1139 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑧𝐵)
1918, 16eleqtrd 2843 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑧 ∈ (Base‘𝐶))
206, 7, 11, 13, 17, 19homfeqval 17740 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((2nd𝑢)𝐻𝑧) = ((2nd𝑢)(Hom ‘𝐷)𝑧))
21 xp1st 8046 . . . . . . . . . . . 12 (𝑢 ∈ (𝐵 × 𝐵) → (1st𝑢) ∈ 𝐵)
22213ad2ant2 1135 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (1st𝑢) ∈ 𝐵)
2322, 16eleqtrd 2843 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (1st𝑢) ∈ (Base‘𝐶))
246, 7, 11, 13, 23, 17homfeqval 17740 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((1st𝑢)𝐻(2nd𝑢)) = ((1st𝑢)(Hom ‘𝐷)(2nd𝑢)))
25 df-ov 7434 . . . . . . . . 9 ((1st𝑢)𝐻(2nd𝑢)) = (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩)
26 df-ov 7434 . . . . . . . . 9 ((1st𝑢)(Hom ‘𝐷)(2nd𝑢)) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩)
2724, 25, 263eqtr3g 2800 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩))
28 1st2nd2 8053 . . . . . . . . . 10 (𝑢 ∈ (𝐵 × 𝐵) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
29283ad2ant2 1135 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
3029fveq2d 6910 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻𝑢) = (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩))
3129fveq2d 6910 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((Hom ‘𝐷)‘𝑢) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩))
3227, 30, 313eqtr4d 2787 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻𝑢) = ((Hom ‘𝐷)‘𝑢))
33 eqidd 2738 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝑔(𝑢 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓))
3420, 32, 33mpoeq123dv 7508 . . . . . 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)))
3534mpoeq3dva 7510 . . . . 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))))
36 comfeq.4 . . . . . . 7 (𝜑𝐵 = (Base‘𝐷))
3736sqxpeqd 5717 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐷) × (Base‘𝐷)))
38 eqidd 2738 . . . . . 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)))
3937, 36, 38mpoeq123dv 7508 . . . . 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))))
4035, 39eqtrd 2777 . . . 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))))
41 eqid 2737 . . . . 5 (compf𝐷) = (compf𝐷)
42 eqid 2737 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
43 comfeq.2 . . . . 5 = (comp‘𝐷)
4441, 42, 11, 43comfffval 17741 . . . 4 (compf𝐷) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ ((2nd𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 𝑧)𝑓)))
4540, 44eqtr4di 2795 . . 3 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) = (compf𝐷))
4610, 45eqeq12d 2753 . 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)))
47 ovex 7464 . . . . . 6 ((2nd𝑢)𝐻𝑧) ∈ V
48 fvex 6919 . . . . . 6 (𝐻𝑢) ∈ V
4947, 48mpoex 8104 . . . . 5 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
5049rgen2w 3066 . . . 4 𝑢 ∈ (𝐵 × 𝐵)∀𝑧𝐵 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
51 mpo2eqb 7565 . . . 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))))
5250, 51ax-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)))
53 vex 3484 . . . . . . . . 9 𝑥 ∈ V
54 vex 3484 . . . . . . . . 9 𝑦 ∈ V
5553, 54op2ndd 8025 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (2nd𝑢) = 𝑦)
5655oveq1d 7446 . . . . . . 7 (𝑢 = ⟨𝑥, 𝑦⟩ → ((2nd𝑢)𝐻𝑧) = (𝑦𝐻𝑧))
57 fveq2 6906 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻𝑢) = (𝐻‘⟨𝑥, 𝑦⟩))
58 df-ov 7434 . . . . . . . . 9 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
5957, 58eqtr4di 2795 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻𝑢) = (𝑥𝐻𝑦))
60 oveq1 7438 . . . . . . . . . 10 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 · 𝑧) = (⟨𝑥, 𝑦· 𝑧))
6160oveqd 7448 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))
62 oveq1 7438 . . . . . . . . . 10 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 𝑧) = (⟨𝑥, 𝑦 𝑧))
6362oveqd 7448 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓))
6461, 63eqeq12d 2753 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓) ↔ (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
6559, 64raleqbidv 3346 . . . . . . 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)))
6656, 65raleqbidv 3346 . . . . . 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)))
67 ovex 7464 . . . . . . . 8 (𝑔(𝑢 · 𝑧)𝑓) ∈ V
6867rgen2w 3066 . . . . . . 7 𝑔 ∈ ((2nd𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻𝑢)(𝑔(𝑢 · 𝑧)𝑓) ∈ V
69 mpo2eqb 7565 . . . . . . 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)))
7068, 69ax-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))
71 ralcom 3289 . . . . . 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))
7266, 70, 713bitr4g 314 . . . . 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)))
7372ralbidv 3178 . . . 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)))
7473ralxp 5852 . . 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))
7552, 74bitri 275 . 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))
7646, 75bitr3di 286 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 206  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cop 4632   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  compcco 17309  Homf chomf 17709  compfccomf 17710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-homf 17713  df-comf 17714
This theorem is referenced by:  comfeqd  17750  2oppccomf  17768  oppccomfpropd  17770  resssetc  18137  resscatc  18154  oppcthinco  49088  oppcthinendcALT  49090
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