Theorem rngccatidALTV 44259

Description: Lemma for rngccatALTV 44260. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)

Hypotheses

Ref	Expression
rngccatALTV.c	⊢ 𝐶 = (RngCatALTV‘𝑈)
rngccatidALTV.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
rngccatidALTV	⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝑈   𝑥,𝑉

Proof of Theorem rngccatidALTV

Dummy variables 𝑓 𝑔 ℎ 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngccatidALTV.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2	1	a1i 11	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐵 = (Base‘𝐶))
3		eqidd 2822	. 2 ⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		eqidd 2822	. 2 ⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶))
5		rngccatALTV.c	. . . 4 ⊢ 𝐶 = (RngCatALTV‘𝑈)
6	5	fvexi 6683	. . 3 ⊢ 𝐶 ∈ V
7	6	a1i 11	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
8		biid 263	. 2 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) ↔ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))))
9		simpl 485	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉)
10	5, 1, 9	rngcbasALTV 44253	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (𝑈 ∩ Rng))
11		eleq2 2901	. . . . . . . 8 ⊢ (𝐵 = (𝑈 ∩ Rng) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng)))
12		elin 4168	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng))
13	12	simprbi 499	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥 ∈ Rng)
14	11, 13	syl6bi 255	. . . . . . 7 ⊢ (𝐵 = (𝑈 ∩ Rng) → (𝑥 ∈ 𝐵 → 𝑥 ∈ Rng))
15	14	com12 32	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
16	15	adantl 484	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
17	10, 16	mpd 15	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Rng)
18		eqid 2821	. . . . 5 ⊢ (Base‘𝑥) = (Base‘𝑥)
19	18	idrnghm 44178	. . . 4 ⊢ (𝑥 ∈ Rng → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
20	17, 19	syl 17	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
21		eqid 2821	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
22		simpr 487	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
23	5, 1, 9, 21, 22, 22	rngchomALTV 44255	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RngHomo 𝑥))
24	20, 23	eleqtrrd 2916	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
25		simpl 485	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈 ∈ 𝑉)
26		eqid 2821	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
27		simpl 485	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑤 ∈ 𝐵)
28	27	3ad2ant1 1129	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑤 ∈ 𝐵)
29	28	adantl 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤 ∈ 𝐵)
30		simpr 487	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
31	30	3ad2ant1 1129	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ 𝐵)
32	31	adantl 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥 ∈ 𝐵)
33		simp1 1132	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑈 ∈ 𝑉)
34	27	3ad2ant3 1131	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
35	30	3ad2ant3 1131	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
36	5, 1, 33, 21, 34, 35	rngchomALTV 44255	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RngHomo 𝑥))
37	36	eleq2d 2898	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓 ∈ (𝑤 RngHomo 𝑥)))
38	37	biimpd 231	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RngHomo 𝑥)))
39	38	3exp 1115	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RngHomo 𝑥)))))
40	39	com14 96	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑓 ∈ (𝑤 RngHomo 𝑥)))))
41	40	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑓 ∈ (𝑤 RngHomo 𝑥)))))
42	41	com13 88	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → 𝑓 ∈ (𝑤 RngHomo 𝑥)))))
43	42	3imp 1107	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → 𝑓 ∈ (𝑤 RngHomo 𝑥)))
44	43	impcom 410	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤 RngHomo 𝑥))
45	20	expcom 416	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
46	45	adantl 484	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
47	46	3ad2ant1 1129	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
48	47	impcom 410	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
49	5, 1, 25, 26, 29, 32, 32, 44, 48	rngccoALTV 44258	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = (( I ↾ (Base‘𝑥)) ∘ 𝑓))
50		simpl 485	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑈 ∈ 𝑉)
51		simprl 769	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
52		simprr 771	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
53	5, 1, 50, 21, 51, 52	elrngchomALTV 44256	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥)))
54	53	ex 415	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑉 → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥))))
55	54	com13 88	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑓:(Base‘𝑤)⟶(Base‘𝑥))))
56		fcoi2 6552	. . . . . . . . 9 ⊢ (𝑓:(Base‘𝑤)⟶(Base‘𝑥) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
57	55, 56	syl8 76	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
58	57	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
59	58	com12 32	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
60	59	a1d 25	. . . . 5 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓))))
61	60	3imp 1107	. . . 4 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓))
62	61	impcom 410	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
63	49, 62	eqtrd 2856	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
64		simp3 1134	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑈 ∈ 𝑉)
65	30	adantr 483	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
66	65	3ad2ant2 1130	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑥 ∈ 𝐵)
67		simprl 769	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
68	67	3ad2ant2 1130	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑦 ∈ 𝐵)
69	46	adantr 483	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
70	69	a1i 11	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))))
71	70	3imp 1107	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
72		simpl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → 𝑈 ∈ 𝑉)
73	65	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → 𝑥 ∈ 𝐵)
74	67	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → 𝑦 ∈ 𝐵)
75	5, 1, 72, 21, 73, 74	rngchomALTV 44255	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHomo 𝑦))
76	75	eleq2d 2898	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHomo 𝑦)))
77	76	biimpd 231	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))
78	77	ex 415	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝑉 → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦))))
79	78	com13 88	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RngHomo 𝑦))))
80	79	3imp 1107	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑔 ∈ (𝑥 RngHomo 𝑦))
81	5, 1, 64, 26, 66, 66, 68, 71, 80	rngccoALTV 44258	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = (𝑔 ∘ ( I ↾ (Base‘𝑥))))
82	5, 1, 72, 21, 73, 74	elrngchomALTV 44256	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦)))
83	82	ex 415	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝑉 → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))))
84	83	com13 88	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))))
85	84	3imp 1107	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
86		fcoi1 6551	. . . . . . . . 9 ⊢ (𝑔:(Base‘𝑥)⟶(Base‘𝑦) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
87	85, 86	syl 17	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
88	81, 87	eqtrd 2856	. . . . . . 7 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
89	88	3exp 1115	. . . . . 6 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
90	89	3ad2ant2 1130	. . . . 5 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑈 ∈ 𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
91	90	expdcom 417	. . . 4 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔))))
92	91	3imp 1107	. . 3 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔))
93	92	impcom 410	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
94		simp2l 1195	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
95	5, 1, 33, 21, 35, 94	rngchomALTV 44255	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHomo 𝑦))
96	95	eleq2d 2898	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHomo 𝑦)))
97	96	biimpd 231	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))
98	97	3exp 1115	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑉 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))))
99	98	com14 96	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RngHomo 𝑦)))))
100	99	3ad2ant2 1130	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RngHomo 𝑦)))))
101	100	com13 88	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RngHomo 𝑦)))))
102	101	3imp 1107	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → 𝑔 ∈ (𝑥 RngHomo 𝑦)))
103	102	impcom 410	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥 RngHomo 𝑦))
104		rnghmco 44177	. . . 4 ⊢ ((𝑔 ∈ (𝑥 RngHomo 𝑦) ∧ 𝑓 ∈ (𝑤 RngHomo 𝑥)) → (𝑔 ∘ 𝑓) ∈ (𝑤 RngHomo 𝑦))
105	103, 44, 104	syl2anc 586	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ 𝑓) ∈ (𝑤 RngHomo 𝑦))
106		simp2l 1195	. . . . 5 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ 𝐵)
107	106	adantl 484	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦 ∈ 𝐵)
108	5, 1, 25, 26, 29, 32, 107, 44, 103	rngccoALTV 44258	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) = (𝑔 ∘ 𝑓))
109	5, 1, 25, 21, 29, 107	rngchomALTV 44255	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑦) = (𝑤 RngHomo 𝑦))
110	105, 108, 109	3eltr4d 2928	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦))
111		coass 6117	. . . 4 ⊢ ((ℎ ∘ 𝑔) ∘ 𝑓) = (ℎ ∘ (𝑔 ∘ 𝑓))
112		simp2r 1196	. . . . . 6 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ 𝐵)
113	112	adantl 484	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧 ∈ 𝐵)
114		simp2r 1196	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
115	5, 1, 33, 21, 94, 114	rngchomALTV 44255	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RngHomo 𝑧))
116	115	eleq2d 2898	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ ℎ ∈ (𝑦 RngHomo 𝑧)))
117	116	biimpd 231	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) → ℎ ∈ (𝑦 RngHomo 𝑧)))
118	117	3exp 1115	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝑉 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) → ℎ ∈ (𝑦 RngHomo 𝑧)))))
119	118	com14 96	. . . . . . . . . 10 ⊢ (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → ℎ ∈ (𝑦 RngHomo 𝑧)))))
120	119	3ad2ant3 1131	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → ℎ ∈ (𝑦 RngHomo 𝑧)))))
121	120	com13 88	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈 ∈ 𝑉 → ℎ ∈ (𝑦 RngHomo 𝑧)))))
122	121	3imp 1107	. . . . . . 7 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈 ∈ 𝑉 → ℎ ∈ (𝑦 RngHomo 𝑧)))
123	122	impcom 410	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ℎ ∈ (𝑦 RngHomo 𝑧))
124		rnghmco 44177	. . . . . 6 ⊢ ((ℎ ∈ (𝑦 RngHomo 𝑧) ∧ 𝑔 ∈ (𝑥 RngHomo 𝑦)) → (ℎ ∘ 𝑔) ∈ (𝑥 RngHomo 𝑧))
125	123, 103, 124	syl2anc 586	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ ∘ 𝑔) ∈ (𝑥 RngHomo 𝑧))
126	5, 1, 25, 26, 29, 32, 113, 44, 125	rngccoALTV 44258	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘ 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((ℎ ∘ 𝑔) ∘ 𝑓))
127	5, 1, 25, 26, 29, 107, 113, 105, 123	rngccoALTV 44258	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘ 𝑓)) = (ℎ ∘ (𝑔 ∘ 𝑓)))
128	111, 126, 127	3eqtr4a 2882	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ ∘ 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘ 𝑓)))
129	5, 1, 25, 26, 32, 107, 113, 103, 123	rngccoALTV 44258	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) = (ℎ ∘ 𝑔))
130	129	oveq1d 7170	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((ℎ ∘ 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓))
131	108	oveq2d 7171	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔 ∘ 𝑓)))
132	128, 130, 131	3eqtr4d 2866	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)))
133	2, 3, 4, 7, 8, 24, 63, 93, 110, 132	iscatd2 16951	1 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∩ cin 3934  ⟨cop 4572   ↦ cmpt 5145   I cid 5458   ↾ cres 5556   ∘ ccom 5558  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  Hom chom 16575  compcco 16576  Catccat 16934  Idccid 16935  Rngcrng 44144   RngHomo crngh 44155  RngCatALTVcrngcALTV 44228

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-fz 12892  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-plusg 16577  df-hom 16588  df-cco 16589  df-0g 16714  df-cat 16938  df-cid 16939  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-mhm 17955  df-grp 18105  df-ghm 18355  df-abl 18908  df-mgp 19239  df-mgmhm 44045  df-rng0 44145  df-rnghomo 44157  df-rngcALTV 44230

This theorem is referenced by:  rngccatALTV  44260  rngcidALTV  44261  rhmsubcALTVlem3  44376