Theorem comfeq 17764

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	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
comfeq.4	⊢ (𝜑 → 𝐵 = (Base‘𝐷))
comfeq.5	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))

Assertion

Ref	Expression
comfeq	⊢ (𝜑 → ((compf‘𝐶) = (compf‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (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.

Step	Hyp	Ref	Expression
1		comfeq.3	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
2	1	sqxpeqd 5732	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
3		eqidd 2741	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)))
4	2, 1, 3	mpoeq123dv 7525	. . . 4 ⊢ (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))))
5		eqid 2740	. . . . 5 ⊢ (compf‘𝐶) = (compf‘𝐶)
6		eqid 2740	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		comfeq.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
8		comfeq.1	. . . . 5 ⊢ · = (comp‘𝐶)
9	5, 6, 7, 8	comfffval 17756	. . . 4 ⊢ (compf‘𝐶) = (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)))
10	4, 9	eqtr4di 2798	. . 3 ⊢ (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (compf‘𝐶))
11		eqid 2740	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
12		comfeq.5	. . . . . . . . 9 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
13	12	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (Homf ‘𝐶) = (Homf ‘𝐷))
14		xp2nd 8063	. . . . . . . . . 10 ⊢ (𝑢 ∈ (𝐵 × 𝐵) → (2nd ‘𝑢) ∈ 𝐵)
15	14	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (2nd ‘𝑢) ∈ 𝐵)
16	1	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝐵 = (Base‘𝐶))
17	15, 16	eleqtrd 2846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (2nd ‘𝑢) ∈ (Base‘𝐶))
18		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
19	18, 16	eleqtrd 2846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (Base‘𝐶))
20	6, 7, 11, 13, 17, 19	homfeqval 17755	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → ((2nd ‘𝑢)𝐻𝑧) = ((2nd ‘𝑢)(Hom ‘𝐷)𝑧))
21		xp1st 8062	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (𝐵 × 𝐵) → (1st ‘𝑢) ∈ 𝐵)
22	21	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (1st ‘𝑢) ∈ 𝐵)
23	22, 16	eleqtrd 2846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (1st ‘𝑢) ∈ (Base‘𝐶))
24	6, 7, 11, 13, 23, 17	homfeqval 17755	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → ((1st ‘𝑢)𝐻(2nd ‘𝑢)) = ((1st ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑢)))
25		df-ov 7451	. . . . . . . . 9 ⊢ ((1st ‘𝑢)𝐻(2nd ‘𝑢)) = (𝐻‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
26		df-ov 7451	. . . . . . . . 9 ⊢ ((1st ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑢)) = ((Hom ‘𝐷)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
27	24, 25, 26	3eqtr3g 2803	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐻‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩) = ((Hom ‘𝐷)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩))
28		1st2nd2 8069	. . . . . . . . . 10 ⊢ (𝑢 ∈ (𝐵 × 𝐵) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
29	28	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
30	29	fveq2d 6924	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐻‘𝑢) = (𝐻‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩))
31	29	fveq2d 6924	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → ((Hom ‘𝐷)‘𝑢) = ((Hom ‘𝐷)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩))
32	27, 30, 31	3eqtr4d 2790	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐻‘𝑢) = ((Hom ‘𝐷)‘𝑢))
33		eqidd 2741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝑔(𝑢 ∙ 𝑧)𝑓) = (𝑔(𝑢 ∙ 𝑧)𝑓))
34	20, 32, 33	mpoeq123dv 7525	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)))
35	34	mpoeq3dva 7527	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))))
36		comfeq.4	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
37	36	sqxpeqd 5732	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐷) × (Base‘𝐷)))
38		eqidd 2741	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ ((2nd ‘𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)))
39	37, 36, 38	mpoeq123dv 7525	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ ((2nd ‘𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))))
40	35, 39	eqtrd 2780	. . . 4 ⊢ (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ ((2nd ‘𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))))
41		eqid 2740	. . . . 5 ⊢ (compf‘𝐷) = (compf‘𝐷)
42		eqid 2740	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
43		comfeq.2	. . . . 5 ⊢ ∙ = (comp‘𝐷)
44	41, 42, 11, 43	comfffval 17756	. . . 4 ⊢ (compf‘𝐷) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐷)), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ ((2nd ‘𝑢)(Hom ‘𝐷)𝑧), 𝑓 ∈ ((Hom ‘𝐷)‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)))
45	40, 44	eqtr4di 2798	. . 3 ⊢ (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))) = (compf‘𝐷))
46	10, 45	eqeq12d 2756	. 2 ⊢ (𝜑 → ((𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))) ↔ (compf‘𝐶) = (compf‘𝐷)))
47		ovex 7481	. . . . . 6 ⊢ ((2nd ‘𝑢)𝐻𝑧) ∈ V
48		fvex 6933	. . . . . 6 ⊢ (𝐻‘𝑢) ∈ V
49	47, 48	mpoex 8120	. . . . 5 ⊢ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
50	49	rgen2w 3072	. . . 4 ⊢ ∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧 ∈ 𝐵 (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
51		mpo2eqb 7582	. . . 4 ⊢ (∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧 ∈ 𝐵 (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V → ((𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))) ↔ ∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧 ∈ 𝐵 (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))))
52	50, 51	ax-mp 5	. . 3 ⊢ ((𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))) ↔ ∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧 ∈ 𝐵 (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)))
53		vex 3492	. . . . . . . . 9 ⊢ 𝑥 ∈ V
54		vex 3492	. . . . . . . . 9 ⊢ 𝑦 ∈ V
55	53, 54	op2ndd 8041	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑢) = 𝑦)
56	55	oveq1d 7463	. . . . . . 7 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((2nd ‘𝑢)𝐻𝑧) = (𝑦𝐻𝑧))
57		fveq2 6920	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑢) = (𝐻‘⟨𝑥, 𝑦⟩))
58		df-ov 7451	. . . . . . . . 9 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
59	57, 58	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑢) = (𝑥𝐻𝑦))
60		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 · 𝑧) = (⟨𝑥, 𝑦⟩ · 𝑧))
61	60	oveqd 7465	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))
62		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 ∙ 𝑧) = (⟨𝑥, 𝑦⟩ ∙ 𝑧))
63	62	oveqd 7465	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 ∙ 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓))
64	61, 63	eqeq12d 2756	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 ∙ 𝑧)𝑓) ↔ (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓)))
65	59, 64	raleqbidv 3354	. . . . . . 7 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑓 ∈ (𝐻‘𝑢)(𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 ∙ 𝑧)𝑓) ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓)))
66	56, 65	raleqbidv 3354	. . . . . 6 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻‘𝑢)(𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 ∙ 𝑧)𝑓) ↔ ∀𝑔 ∈ (𝑦𝐻𝑧)∀𝑓 ∈ (𝑥𝐻𝑦)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓)))
67		ovex 7481	. . . . . . . 8 ⊢ (𝑔(𝑢 · 𝑧)𝑓) ∈ V
68	67	rgen2w 3072	. . . . . . 7 ⊢ ∀𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻‘𝑢)(𝑔(𝑢 · 𝑧)𝑓) ∈ V
69		mpo2eqb 7582	. . . . . . 7 ⊢ (∀𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻‘𝑢)(𝑔(𝑢 · 𝑧)𝑓) ∈ V → ((𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)) ↔ ∀𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻‘𝑢)(𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 ∙ 𝑧)𝑓)))
70	68, 69	ax-mp 5	. . . . . 6 ⊢ ((𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)) ↔ ∀𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻‘𝑢)(𝑔(𝑢 · 𝑧)𝑓) = (𝑔(𝑢 ∙ 𝑧)𝑓))
71		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))
72	66, 70, 71	3bitr4g 314	. . . . 5 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)) ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓)))
73	72	ralbidv 3184	. . . 4 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑧 ∈ 𝐵 (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)) ↔ ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓)))
74	73	ralxp 5866	. . 3 ⊢ (∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧 ∈ 𝐵 (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) = (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓))
75	52, 74	bitri 275	. 2 ⊢ ((𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓))
76	46, 75	bitr3di 286	1 ⊢ (𝜑 → ((compf‘𝐶) = (compf‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  ⟨cop 4654   × cxp 5698  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029  Basecbs 17258  Hom chom 17322  compcco 17323  Hom_f chomf 17724  comp_fccomf 17725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-homf 17728  df-comf 17729

This theorem is referenced by:  comfeqd  17765  2oppccomf  17785  oppccomfpropd  17787  resssetc  18159  resscatc  18176