Theorem rngccatidALTV 45174

Description: Lemma for rngccatALTV 45175. (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 2735	. 2 ⊢ (𝑈 ∈ 𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4		eqidd 2735	. 2 ⊢ (𝑈 ∈ 𝑉 → (comp‘𝐶) = (comp‘𝐶))
5		rngccatALTV.c	. . . 4 ⊢ 𝐶 = (RngCatALTV‘𝑈)
6	5	fvexi 6720	. . 3 ⊢ 𝐶 ∈ V
7	6	a1i 11	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
8		biid 264	. 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 486	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉)
10	5, 1, 9	rngcbasALTV 45168	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (𝑈 ∩ Rng))
11		eleq2 2822	. . . . . . . 8 ⊢ (𝐵 = (𝑈 ∩ Rng) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng)))
12		elin 3873	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng))
13	12	simprbi 500	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥 ∈ Rng)
14	11, 13	syl6bi 256	. . . . . . 7 ⊢ (𝐵 = (𝑈 ∩ Rng) → (𝑥 ∈ 𝐵 → 𝑥 ∈ Rng))
15	14	com12 32	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
16	15	adantl 485	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
17	10, 16	mpd 15	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Rng)
18		eqid 2734	. . . . 5 ⊢ (Base‘𝑥) = (Base‘𝑥)
19	18	idrnghm 45093	. . . 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 2734	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
22		simpr 488	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
23	5, 1, 9, 21, 22, 22	rngchomALTV 45170	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RngHomo 𝑥))
24	20, 23	eleqtrrd 2837	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
25		simpl 486	. . . 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 2734	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
27		simpl 486	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑤 ∈ 𝐵)
28	27	3ad2ant1 1135	. . . . 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 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)))) → 𝑤 ∈ 𝐵)
30		simpr 488	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
31	30	3ad2ant1 1135	. . . . 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 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)))) → 𝑥 ∈ 𝐵)
33		simp1 1138	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑈 ∈ 𝑉)
34	27	3ad2ant3 1137	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
35	30	3ad2ant3 1137	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
36	5, 1, 33, 21, 34, 35	rngchomALTV 45170	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RngHomo 𝑥))
37	36	eleq2d 2819	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓 ∈ (𝑤 RngHomo 𝑥)))
38	37	biimpd 232	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RngHomo 𝑥)))
39	38	3exp 1121	. . . . . . . . 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 1135	. . . . . . 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 1113	. . . . 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 411	. . . 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 417	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
46	45	adantl 485	. . . . . 6 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑈 ∈ 𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
47	46	3ad2ant1 1135	. . . . 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 411	. . . 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 45173	. . 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 486	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑈 ∈ 𝑉)
51		simprl 771	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
52		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
53	5, 1, 50, 21, 51, 52	elrngchomALTV 45171	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥)))
54	53	ex 416	. . . . . . . . . 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 6583	. . . . . . . . 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 1135	. . . . . . 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 1113	. . . 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 411	. . 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 2774	. 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 1140	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑈 ∈ 𝑉)
65	30	adantr 484	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
66	65	3ad2ant2 1136	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑥 ∈ 𝐵)
67		simprl 771	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
68	67	3ad2ant2 1136	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑦 ∈ 𝐵)
69	46	adantr 484	. . . . . . . . . . 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 1113	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
72		simpl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → 𝑈 ∈ 𝑉)
73	65	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → 𝑥 ∈ 𝐵)
74	67	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → 𝑦 ∈ 𝐵)
75	5, 1, 72, 21, 73, 74	rngchomALTV 45170	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHomo 𝑦))
76	75	eleq2d 2819	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHomo 𝑦)))
77	76	biimpd 232	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))
78	77	ex 416	. . . . . . . . . . 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 1113	. . . . . . . . 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 45173	. . . . . . . 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 45171	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦)))
83	82	ex 416	. . . . . . . . . . 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 1113	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑈 ∈ 𝑉) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
86		fcoi1 6582	. . . . . . . . 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 2774	. . . . . . 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 1121	. . . . . 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 1136	. . . . 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 418	. . . 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 1113	. . 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 411	. 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 1201	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
95	5, 1, 33, 21, 35, 94	rngchomALTV 45170	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHomo 𝑦))
96	95	eleq2d 2819	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHomo 𝑦)))
97	96	biimpd 232	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))
98	97	3exp 1121	. . . . . . . . 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 1136	. . . . . . 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 1113	. . . . 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 411	. . . 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 45092	. . . 4 ⊢ ((𝑔 ∈ (𝑥 RngHomo 𝑦) ∧ 𝑓 ∈ (𝑤 RngHomo 𝑥)) → (𝑔 ∘ 𝑓) ∈ (𝑤 RngHomo 𝑦))
105	103, 44, 104	syl2anc 587	. . 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 1201	. . . . 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 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)))) → 𝑦 ∈ 𝐵)
108	5, 1, 25, 26, 29, 32, 107, 44, 103	rngccoALTV 45173	. . 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 45170	. . 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 2849	. 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 6118	. . . 4 ⊢ ((ℎ ∘ 𝑔) ∘ 𝑓) = (ℎ ∘ (𝑔 ∘ 𝑓))
112		simp2r 1202	. . . . . 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 485	. . . . 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 1202	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
115	5, 1, 33, 21, 94, 114	rngchomALTV 45170	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RngHomo 𝑧))
116	115	eleq2d 2819	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ ℎ ∈ (𝑦 RngHomo 𝑧)))
117	116	biimpd 232	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (ℎ ∈ (𝑦(Hom ‘𝐶)𝑧) → ℎ ∈ (𝑦 RngHomo 𝑧)))
118	117	3exp 1121	. . . . . . . . . . 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 1137	. . . . . . . . 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 1113	. . . . . . 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 411	. . . . . 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 45092	. . . . . 6 ⊢ ((ℎ ∈ (𝑦 RngHomo 𝑧) ∧ 𝑔 ∈ (𝑥 RngHomo 𝑦)) → (ℎ ∘ 𝑔) ∈ (𝑥 RngHomo 𝑧))
125	123, 103, 124	syl2anc 587	. . . . 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 45173	. . . 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 45173	. . . 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 2800	. . 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 45173	. . . 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 7217	. . 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 7218	. . 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 2784	. 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 17156	1 ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  Vcvv 3401   ∩ cin 3856  ⟨cop 4537   ↦ cmpt 5124   I cid 5443   ↾ cres 5542   ∘ ccom 5544  ⟶wf 6365  ‘cfv 6369  (class class class)co 7202  Basecbs 16684  Hom chom 16778  compcco 16779  Catccat 17139  Idccid 17140  Rngcrng 45059   RngHomo crngh 45070  RngCatALTVcrngcALTV 45143

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 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-er 8380  df-map 8499  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-2 11876  df-3 11877  df-4 11878  df-5 11879  df-6 11880  df-7 11881  df-8 11882  df-9 11883  df-n0 12074  df-z 12160  df-dec 12277  df-uz 12422  df-fz 13079  df-struct 16686  df-ndx 16687  df-slot 16688  df-base 16690  df-sets 16691  df-plusg 16780  df-hom 16791  df-cco 16792  df-0g 16918  df-cat 17143  df-cid 17144  df-mgm 18086  df-sgrp 18135  df-mnd 18146  df-mhm 18190  df-grp 18340  df-ghm 18592  df-abl 19145  df-mgp 19477  df-mgmhm 44960  df-rng0 45060  df-rnghomo 45072  df-rngcALTV 45145

This theorem is referenced by:  rngccatALTV  45175  rngcidALTV  45176  rhmsubcALTVlem3  45291