Theorem comfeq 17209

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 5583	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶)))
3		eqidd 2738	. . . . 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 7286	. . . 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 2737	. . . . 5 ⊢ (compf‘𝐶) = (compf‘𝐶)
6		eqid 2737	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		comfeq.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
8		comfeq.1	. . . . 5 ⊢ · = (comp‘𝐶)
9	5, 6, 7, 8	comfffval 17201	. . . 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 2796	. . 3 ⊢ (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓))) = (compf‘𝐶))
11		eqid 2737	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
12		comfeq.5	. . . . . . . . 9 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
13	12	3ad2ant1 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (Homf ‘𝐶) = (Homf ‘𝐷))
14		xp2nd 7794	. . . . . . . . . 10 ⊢ (𝑢 ∈ (𝐵 × 𝐵) → (2nd ‘𝑢) ∈ 𝐵)
15	14	3ad2ant2 1136	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (2nd ‘𝑢) ∈ 𝐵)
16	1	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝐵 = (Base‘𝐶))
17	15, 16	eleqtrd 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (2nd ‘𝑢) ∈ (Base‘𝐶))
18		simp3 1140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
19	18, 16	eleqtrd 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (Base‘𝐶))
20	6, 7, 11, 13, 17, 19	homfeqval 17200	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → ((2nd ‘𝑢)𝐻𝑧) = ((2nd ‘𝑢)(Hom ‘𝐷)𝑧))
21		xp1st 7793	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (𝐵 × 𝐵) → (1st ‘𝑢) ∈ 𝐵)
22	21	3ad2ant2 1136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (1st ‘𝑢) ∈ 𝐵)
23	22, 16	eleqtrd 2840	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (1st ‘𝑢) ∈ (Base‘𝐶))
24	6, 7, 11, 13, 23, 17	homfeqval 17200	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → ((1st ‘𝑢)𝐻(2nd ‘𝑢)) = ((1st ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑢)))
25		df-ov 7216	. . . . . . . . 9 ⊢ ((1st ‘𝑢)𝐻(2nd ‘𝑢)) = (𝐻‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
26		df-ov 7216	. . . . . . . . 9 ⊢ ((1st ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑢)) = ((Hom ‘𝐷)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
27	24, 25, 26	3eqtr3g 2801	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐻‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩) = ((Hom ‘𝐷)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩))
28		1st2nd2 7800	. . . . . . . . . 10 ⊢ (𝑢 ∈ (𝐵 × 𝐵) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
29	28	3ad2ant2 1136	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
30	29	fveq2d 6721	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐻‘𝑢) = (𝐻‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩))
31	29	fveq2d 6721	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → ((Hom ‘𝐷)‘𝑢) = ((Hom ‘𝐷)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩))
32	27, 30, 31	3eqtr4d 2787	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐻‘𝑢) = ((Hom ‘𝐷)‘𝑢))
33		eqidd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐵 × 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝑔(𝑢 ∙ 𝑧)𝑓) = (𝑔(𝑢 ∙ 𝑧)𝑓))
34	20, 32, 33	mpoeq123dv 7286	. . . . . 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 7288	. . . . 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 5583	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐷) × (Base‘𝐷)))
38		eqidd 2738	. . . . . 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 7286	. . . . 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 2777	. . . 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 2737	. . . . 5 ⊢ (compf‘𝐷) = (compf‘𝐷)
42		eqid 2737	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
43		comfeq.2	. . . . 5 ⊢ ∙ = (comp‘𝐷)
44	41, 42, 11, 43	comfffval 17201	. . . 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 2796	. . 3 ⊢ (𝜑 → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 ∙ 𝑧)𝑓))) = (compf‘𝐷))
46	10, 45	eqeq12d 2753	. 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 7246	. . . . . 6 ⊢ ((2nd ‘𝑢)𝐻𝑧) ∈ V
48		fvex 6730	. . . . . 6 ⊢ (𝐻‘𝑢) ∈ V
49	47, 48	mpoex 7850	. . . . 5 ⊢ (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
50	49	rgen2w 3074	. . . 4 ⊢ ∀𝑢 ∈ (𝐵 × 𝐵)∀𝑧 ∈ 𝐵 (𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑢) ↦ (𝑔(𝑢 · 𝑧)𝑓)) ∈ V
51		mpo2eqb 7342	. . . 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 3412	. . . . . . . . 9 ⊢ 𝑥 ∈ V
54		vex 3412	. . . . . . . . 9 ⊢ 𝑦 ∈ V
55	53, 54	op2ndd 7772	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑢) = 𝑦)
56	55	oveq1d 7228	. . . . . . 7 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((2nd ‘𝑢)𝐻𝑧) = (𝑦𝐻𝑧))
57		fveq2 6717	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑢) = (𝐻‘⟨𝑥, 𝑦⟩))
58		df-ov 7216	. . . . . . . . 9 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
59	57, 58	eqtr4di 2796	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑢) = (𝑥𝐻𝑦))
60		oveq1 7220	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 · 𝑧) = (⟨𝑥, 𝑦⟩ · 𝑧))
61	60	oveqd 7230	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))
62		oveq1 7220	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑢 ∙ 𝑧) = (⟨𝑥, 𝑦⟩ ∙ 𝑧))
63	62	oveqd 7230	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑔(𝑢 ∙ 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓))
64	61, 63	eqeq12d 2753	. . . . . . . 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 3313	. . . . . . 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 3313	. . . . . 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 7246	. . . . . . . 8 ⊢ (𝑔(𝑢 · 𝑧)𝑓) ∈ V
68	67	rgen2w 3074	. . . . . . 7 ⊢ ∀𝑔 ∈ ((2nd ‘𝑢)𝐻𝑧)∀𝑓 ∈ (𝐻‘𝑢)(𝑔(𝑢 · 𝑧)𝑓) ∈ V
69		mpo2eqb 7342	. . . . . . 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 3267	. . . . . 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 317	. . . . 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 3118	. . . 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 5710	. . 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 278	. 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 289	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 209   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408  ⟨cop 4547   × cxp 5549  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  1^st c1st 7759  2^nd c2nd 7760  Basecbs 16760  Hom chom 16813  compcco 16814  Hom_f chomf 17169  comp_fccomf 17170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-homf 17173  df-comf 17174

This theorem is referenced by:  comfeqd  17210  2oppccomf  17229  oppccomfpropd  17231  resssetc  17598  resscatc  17615