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Theorem comfeq 17751
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 5721 . . . . 5 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
3 eqidd 2736 . . . . 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 2735 . . . . 5 (compf𝐶) = (compf𝐶)
6 eqid 2735 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
7 comfeq.h . . . . 5 𝐻 = (Hom ‘𝐶)
8 comfeq.1 . . . . 5 · = (comp‘𝐶)
95, 6, 7, 8comfffval 17743 . . . 4 (compf𝐶) = (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)))
104, 9eqtr4di 2793 . . 3 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (compf𝐶))
11 eqid 2735 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
12 comfeq.5 . . . . . . . . 9 (𝜑 → (Homf𝐶) = (Homf𝐷))
13123ad2ant1 1132 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (Homf𝐶) = (Homf𝐷))
14 xp2nd 8046 . . . . . . . . . 10 (𝑢 ∈ (𝐵 × 𝐵) → (2nd𝑢) ∈ 𝐵)
15143ad2ant2 1133 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (2nd𝑢) ∈ 𝐵)
1613ad2ant1 1132 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝐵 = (Base‘𝐶))
1715, 16eleqtrd 2841 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (2nd𝑢) ∈ (Base‘𝐶))
18 simp3 1137 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑧𝐵)
1918, 16eleqtrd 2841 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑧 ∈ (Base‘𝐶))
206, 7, 11, 13, 17, 19homfeqval 17742 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((2nd𝑢)𝐻𝑧) = ((2nd𝑢)(Hom ‘𝐷)𝑧))
21 xp1st 8045 . . . . . . . . . . . 12 (𝑢 ∈ (𝐵 × 𝐵) → (1st𝑢) ∈ 𝐵)
22213ad2ant2 1133 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (1st𝑢) ∈ 𝐵)
2322, 16eleqtrd 2841 . . . . . . . . . 10 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (1st𝑢) ∈ (Base‘𝐶))
246, 7, 11, 13, 23, 17homfeqval 17742 . . . . . . . . 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 2798 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩))
28 1st2nd2 8052 . . . . . . . . . 10 (𝑢 ∈ (𝐵 × 𝐵) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
29283ad2ant2 1133 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → 𝑢 = ⟨(1st𝑢), (2nd𝑢)⟩)
3029fveq2d 6911 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻𝑢) = (𝐻‘⟨(1st𝑢), (2nd𝑢)⟩))
3129fveq2d 6911 . . . . . . . 8 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → ((Hom ‘𝐷)‘𝑢) = ((Hom ‘𝐷)‘⟨(1st𝑢), (2nd𝑢)⟩))
3227, 30, 313eqtr4d 2785 . . . . . . 7 ((𝜑𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧𝐵) → (𝐻𝑢) = ((Hom ‘𝐷)‘𝑢))
33 eqidd 2736 . . . . . . 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 5721 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐷) × (Base‘𝐷)))
38 eqidd 2736 . . . . . 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 2775 . . . 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 2735 . . . . 5 (compf𝐷) = (compf𝐷)
42 eqid 2735 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
43 comfeq.2 . . . . 5 = (comp‘𝐷)
4441, 42, 11, 43comfffval 17743 . . . 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 2793 . . 3 (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 𝑧)𝑓))) = (compf𝐷))
4610, 45eqeq12d 2751 . 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 6920 . . . . . 6 (𝐻𝑢) ∈ V
4947, 48mpoex 8103 . . . . 5 (𝑔 ∈ ((2nd𝑢)𝐻𝑧), 𝑓 ∈ (𝐻𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
5049rgen2w 3064 . . . 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 3482 . . . . . . . . 9 𝑥 ∈ V
54 vex 3482 . . . . . . . . 9 𝑦 ∈ V
5553, 54op2ndd 8024 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (2nd𝑢) = 𝑦)
5655oveq1d 7446 . . . . . . 7 (𝑢 = ⟨𝑥, 𝑦⟩ → ((2nd𝑢)𝐻𝑧) = (𝑦𝐻𝑧))
57 fveq2 6907 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻𝑢) = (𝐻‘⟨𝑥, 𝑦⟩))
58 df-ov 7434 . . . . . . . . 9 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
5957, 58eqtr4di 2793 . . . . . . . 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 2751 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 𝑧)𝑓) ↔ (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
6559, 64raleqbidv 3344 . . . . . . 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 3344 . . . . . 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 3064 . . . . . . 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 3287 . . . . . 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 3176 . . . 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 5855 . . 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 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cop 4637   × cxp 5687  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  Hom chom 17309  compcco 17310  Homf chomf 17711  compfccomf 17712
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-homf 17715  df-comf 17716
This theorem is referenced by:  comfeqd  17752  2oppccomf  17772  oppccomfpropd  17774  resssetc  18146  resscatc  18163
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