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Theorem rngccatidALTV 44608
Description: Lemma for rngccatALTV 44609. (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 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝑈   𝑥,𝑉

Proof of Theorem rngccatidALTV
Dummy variables 𝑓 𝑔 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngccatidALTV.b . . 3 𝐵 = (Base‘𝐶)
21a1i 11 . 2 (𝑈𝑉𝐵 = (Base‘𝐶))
3 eqidd 2799 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4 eqidd 2799 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
5 rngccatALTV.c . . . 4 𝐶 = (RngCatALTV‘𝑈)
65fvexi 6659 . . 3 𝐶 ∈ V
76a1i 11 . 2 (𝑈𝑉𝐶 ∈ V)
8 biid 264 . 2 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) ↔ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))))
9 simpl 486 . . . . . 6 ((𝑈𝑉𝑥𝐵) → 𝑈𝑉)
105, 1, 9rngcbasALTV 44602 . . . . 5 ((𝑈𝑉𝑥𝐵) → 𝐵 = (𝑈 ∩ Rng))
11 eleq2 2878 . . . . . . . 8 (𝐵 = (𝑈 ∩ Rng) → (𝑥𝐵𝑥 ∈ (𝑈 ∩ Rng)))
12 elin 3897 . . . . . . . . 9 (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥𝑈𝑥 ∈ Rng))
1312simprbi 500 . . . . . . . 8 (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥 ∈ Rng)
1411, 13syl6bi 256 . . . . . . 7 (𝐵 = (𝑈 ∩ Rng) → (𝑥𝐵𝑥 ∈ Rng))
1514com12 32 . . . . . 6 (𝑥𝐵 → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
1615adantl 485 . . . . 5 ((𝑈𝑉𝑥𝐵) → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
1710, 16mpd 15 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥 ∈ Rng)
18 eqid 2798 . . . . 5 (Base‘𝑥) = (Base‘𝑥)
1918idrnghm 44527 . . . 4 (𝑥 ∈ Rng → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
2017, 19syl 17 . . 3 ((𝑈𝑉𝑥𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
21 eqid 2798 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
22 simpr 488 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥𝐵)
235, 1, 9, 21, 22, 22rngchomALTV 44604 . . 3 ((𝑈𝑉𝑥𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RngHomo 𝑥))
2420, 23eleqtrrd 2893 . 2 ((𝑈𝑉𝑥𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
25 simpl 486 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
26 eqid 2798 . . . 4 (comp‘𝐶) = (comp‘𝐶)
27 simpl 486 . . . . . 6 ((𝑤𝐵𝑥𝐵) → 𝑤𝐵)
28273ad2ant1 1130 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑤𝐵)
2928adantl 485 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤𝐵)
30 simpr 488 . . . . . 6 ((𝑤𝐵𝑥𝐵) → 𝑥𝐵)
31303ad2ant1 1130 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝐵)
3231adantl 485 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥𝐵)
33 simp1 1133 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑈𝑉)
34273ad2ant3 1132 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑤𝐵)
35303ad2ant3 1132 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑥𝐵)
365, 1, 33, 21, 34, 35rngchomALTV 44604 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RngHomo 𝑥))
3736eleq2d 2875 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓 ∈ (𝑤 RngHomo 𝑥)))
3837biimpd 232 . . . . . . . . . 10 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RngHomo 𝑥)))
39383exp 1116 . . . . . . . . 9 (𝑈𝑉 → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RngHomo 𝑥)))))
4039com14 96 . . . . . . . 8 (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑓 ∈ (𝑤 RngHomo 𝑥)))))
41403ad2ant1 1130 . . . . . . 7 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑓 ∈ (𝑤 RngHomo 𝑥)))))
4241com13 88 . . . . . 6 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉𝑓 ∈ (𝑤 RngHomo 𝑥)))))
43423imp 1108 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉𝑓 ∈ (𝑤 RngHomo 𝑥)))
4443impcom 411 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤 RngHomo 𝑥))
4520expcom 417 . . . . . . 7 (𝑥𝐵 → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
4645adantl 485 . . . . . 6 ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
47463ad2ant1 1130 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
4847impcom 411 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
495, 1, 25, 26, 29, 32, 32, 44, 48rngccoALTV 44607 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = (( I ↾ (Base‘𝑥)) ∘ 𝑓))
50 simpl 486 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑈𝑉)
51 simprl 770 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑤𝐵)
52 simprr 772 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑥𝐵)
535, 1, 50, 21, 51, 52elrngchomALTV 44605 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥)))
5453ex 416 . . . . . . . . . 10 (𝑈𝑉 → ((𝑤𝐵𝑥𝐵) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥))))
5554com13 88 . . . . . . . . 9 (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑓:(Base‘𝑤)⟶(Base‘𝑥))))
56 fcoi2 6527 . . . . . . . . 9 (𝑓:(Base‘𝑤)⟶(Base‘𝑥) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
5755, 56syl8 76 . . . . . . . 8 (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
58573ad2ant1 1130 . . . . . . 7 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
5958com12 32 . . . . . 6 ((𝑤𝐵𝑥𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)))
6059a1d 25 . . . . 5 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓))))
61603imp 1108 . . . 4 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓))
6261impcom 411 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
6349, 62eqtrd 2833 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
64 simp3 1135 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑈𝑉)
6530adantr 484 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑥𝐵)
66653ad2ant2 1131 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑥𝐵)
67 simprl 770 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
68673ad2ant2 1131 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑦𝐵)
6946adantr 484 . . . . . . . . . . 11 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
7069a1i 11 . . . . . . . . . 10 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))))
71703imp 1108 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
72 simpl 486 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑈𝑉)
7365adantl 485 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑥𝐵)
7467adantl 485 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑦𝐵)
755, 1, 72, 21, 73, 74rngchomALTV 44604 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHomo 𝑦))
7675eleq2d 2875 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHomo 𝑦)))
7776biimpd 232 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))
7877ex 416 . . . . . . . . . . 11 (𝑈𝑉 → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦))))
7978com13 88 . . . . . . . . . 10 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉𝑔 ∈ (𝑥 RngHomo 𝑦))))
80793imp 1108 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑔 ∈ (𝑥 RngHomo 𝑦))
815, 1, 64, 26, 66, 66, 68, 71, 80rngccoALTV 44607 . . . . . . . 8 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = (𝑔 ∘ ( I ↾ (Base‘𝑥))))
825, 1, 72, 21, 73, 74elrngchomALTV 44605 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦)))
8382ex 416 . . . . . . . . . . 11 (𝑈𝑉 → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))))
8483com13 88 . . . . . . . . . 10 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉𝑔:(Base‘𝑥)⟶(Base‘𝑦))))
85843imp 1108 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
86 fcoi1 6526 . . . . . . . . 9 (𝑔:(Base‘𝑥)⟶(Base‘𝑦) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
8785, 86syl 17 . . . . . . . 8 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
8881, 87eqtrd 2833 . . . . . . 7 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
89883exp 1116 . . . . . 6 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
90893ad2ant2 1131 . . . . 5 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
9190expdcom 418 . . . 4 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔))))
92913imp 1108 . . 3 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔))
9392impcom 411 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
94 simp2l 1196 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑦𝐵)
955, 1, 33, 21, 35, 94rngchomALTV 44604 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHomo 𝑦))
9695eleq2d 2875 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHomo 𝑦)))
9796biimpd 232 . . . . . . . . . 10 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))
98973exp 1116 . . . . . . . . 9 (𝑈𝑉 → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))))
9998com14 96 . . . . . . . 8 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑔 ∈ (𝑥 RngHomo 𝑦)))))
100993ad2ant2 1131 . . . . . . 7 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑔 ∈ (𝑥 RngHomo 𝑦)))))
101100com13 88 . . . . . 6 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉𝑔 ∈ (𝑥 RngHomo 𝑦)))))
1021013imp 1108 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉𝑔 ∈ (𝑥 RngHomo 𝑦)))
103102impcom 411 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥 RngHomo 𝑦))
104 rnghmco 44526 . . . 4 ((𝑔 ∈ (𝑥 RngHomo 𝑦) ∧ 𝑓 ∈ (𝑤 RngHomo 𝑥)) → (𝑔𝑓) ∈ (𝑤 RngHomo 𝑦))
105103, 44, 104syl2anc 587 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔𝑓) ∈ (𝑤 RngHomo 𝑦))
106 simp2l 1196 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
107106adantl 485 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦𝐵)
1085, 1, 25, 26, 29, 32, 107, 44, 103rngccoALTV 44607 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) = (𝑔𝑓))
1095, 1, 25, 21, 29, 107rngchomALTV 44604 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑦) = (𝑤 RngHomo 𝑦))
110105, 108, 1093eltr4d 2905 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦))
111 coass 6085 . . . 4 ((𝑔) ∘ 𝑓) = ( ∘ (𝑔𝑓))
112 simp2r 1197 . . . . . 6 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
113112adantl 485 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧𝐵)
114 simp2r 1197 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑧𝐵)
1155, 1, 33, 21, 94, 114rngchomALTV 44604 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RngHomo 𝑧))
116115eleq2d 2875 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ ∈ (𝑦 RngHomo 𝑧)))
117116biimpd 232 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ∈ (𝑦 RngHomo 𝑧)))
1181173exp 1116 . . . . . . . . . . 11 (𝑈𝑉 → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ∈ (𝑦 RngHomo 𝑧)))))
119118com14 96 . . . . . . . . . 10 ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 ∈ (𝑦 RngHomo 𝑧)))))
1201193ad2ant3 1132 . . . . . . . . 9 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 ∈ (𝑦 RngHomo 𝑧)))))
121120com13 88 . . . . . . . 8 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉 ∈ (𝑦 RngHomo 𝑧)))))
1221213imp 1108 . . . . . . 7 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 ∈ (𝑦 RngHomo 𝑧)))
123122impcom 411 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ∈ (𝑦 RngHomo 𝑧))
124 rnghmco 44526 . . . . . 6 (( ∈ (𝑦 RngHomo 𝑧) ∧ 𝑔 ∈ (𝑥 RngHomo 𝑦)) → (𝑔) ∈ (𝑥 RngHomo 𝑧))
125123, 103, 124syl2anc 587 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔) ∈ (𝑥 RngHomo 𝑧))
1265, 1, 25, 26, 29, 32, 113, 44, 125rngccoALTV 44607 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((𝑔) ∘ 𝑓))
1275, 1, 25, 26, 29, 107, 113, 105, 123rngccoALTV 44607 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔𝑓)) = ( ∘ (𝑔𝑓)))
128111, 126, 1273eqtr4a 2859 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔𝑓)))
1295, 1, 25, 26, 32, 107, 113, 103, 123rngccoALTV 44607 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) = (𝑔))
130129oveq1d 7150 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (((⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓))
131108oveq2d 7151 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)) = ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔𝑓)))
132128, 130, 1313eqtr4d 2843 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (((⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)))
1332, 3, 4, 7, 8, 24, 63, 93, 110, 132iscatd2 16944 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  Vcvv 3441  cin 3880  cop 4531  cmpt 5110   I cid 5424  cres 5521  ccom 5523  wf 6320  cfv 6324  (class class class)co 7135  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928  Rngcrng 44493   RngHomo crngh 44504  RngCatALTVcrngcALTV 44577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-hom 16581  df-cco 16582  df-0g 16707  df-cat 16931  df-cid 16932  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-grp 18098  df-ghm 18348  df-abl 18901  df-mgp 19233  df-mgmhm 44394  df-rng0 44494  df-rnghomo 44506  df-rngcALTV 44579
This theorem is referenced by:  rngccatALTV  44609  rngcidALTV  44610  rhmsubcALTVlem3  44725
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