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Theorem rngccatidALTV 45799
Description: Lemma for rngccatALTV 45800. (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 2738 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4 eqidd 2738 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
5 rngccatALTV.c . . . 4 𝐶 = (RngCatALTV‘𝑈)
65fvexi 6825 . . 3 𝐶 ∈ V
76a1i 11 . 2 (𝑈𝑉𝐶 ∈ V)
8 biid 260 . 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 483 . . . . . 6 ((𝑈𝑉𝑥𝐵) → 𝑈𝑉)
105, 1, 9rngcbasALTV 45793 . . . . 5 ((𝑈𝑉𝑥𝐵) → 𝐵 = (𝑈 ∩ Rng))
11 eleq2 2826 . . . . . . . 8 (𝐵 = (𝑈 ∩ Rng) → (𝑥𝐵𝑥 ∈ (𝑈 ∩ Rng)))
12 elin 3913 . . . . . . . . 9 (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥𝑈𝑥 ∈ Rng))
1312simprbi 497 . . . . . . . 8 (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥 ∈ Rng)
1411, 13syl6bi 252 . . . . . . 7 (𝐵 = (𝑈 ∩ Rng) → (𝑥𝐵𝑥 ∈ Rng))
1514com12 32 . . . . . 6 (𝑥𝐵 → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
1615adantl 482 . . . . 5 ((𝑈𝑉𝑥𝐵) → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
1710, 16mpd 15 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥 ∈ Rng)
18 eqid 2737 . . . . 5 (Base‘𝑥) = (Base‘𝑥)
1918idrnghm 45718 . . . 4 (𝑥 ∈ Rng → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
2017, 19syl 17 . . 3 ((𝑈𝑉𝑥𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
21 eqid 2737 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
22 simpr 485 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥𝐵)
235, 1, 9, 21, 22, 22rngchomALTV 45795 . . 3 ((𝑈𝑉𝑥𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RngHomo 𝑥))
2420, 23eleqtrrd 2841 . 2 ((𝑈𝑉𝑥𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
25 simpl 483 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
26 eqid 2737 . . . 4 (comp‘𝐶) = (comp‘𝐶)
27 simpl 483 . . . . . 6 ((𝑤𝐵𝑥𝐵) → 𝑤𝐵)
28273ad2ant1 1132 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑤𝐵)
2928adantl 482 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤𝐵)
30 simpr 485 . . . . . 6 ((𝑤𝐵𝑥𝐵) → 𝑥𝐵)
31303ad2ant1 1132 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝐵)
3231adantl 482 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥𝐵)
33 simp1 1135 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑈𝑉)
34273ad2ant3 1134 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑤𝐵)
35303ad2ant3 1134 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑥𝐵)
365, 1, 33, 21, 34, 35rngchomALTV 45795 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RngHomo 𝑥))
3736eleq2d 2823 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓 ∈ (𝑤 RngHomo 𝑥)))
3837biimpd 228 . . . . . . . . . 10 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RngHomo 𝑥)))
39383exp 1118 . . . . . . . . 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 1132 . . . . . . 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 1110 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉𝑓 ∈ (𝑤 RngHomo 𝑥)))
4443impcom 408 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤 RngHomo 𝑥))
4520expcom 414 . . . . . . 7 (𝑥𝐵 → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
4645adantl 482 . . . . . 6 ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)))
47463ad2ant1 1132 . . . . 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 408 . . . 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 45798 . . 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 483 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑈𝑉)
51 simprl 768 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑤𝐵)
52 simprr 770 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑥𝐵)
535, 1, 50, 21, 51, 52elrngchomALTV 45796 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥)))
5453ex 413 . . . . . . . . . 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 6686 . . . . . . . . 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 1132 . . . . . . 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 1110 . . . 4 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓))
6261impcom 408 . . 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 2777 . 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 1137 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑈𝑉)
6530adantr 481 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑥𝐵)
66653ad2ant2 1133 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑥𝐵)
67 simprl 768 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
68673ad2ant2 1133 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑦𝐵)
6946adantr 481 . . . . . . . . . . 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 1110 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥))
72 simpl 483 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑈𝑉)
7365adantl 482 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑥𝐵)
7467adantl 482 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑦𝐵)
755, 1, 72, 21, 73, 74rngchomALTV 45795 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHomo 𝑦))
7675eleq2d 2823 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHomo 𝑦)))
7776biimpd 228 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))
7877ex 413 . . . . . . . . . . 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 1110 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑔 ∈ (𝑥 RngHomo 𝑦))
815, 1, 64, 26, 66, 66, 68, 71, 80rngccoALTV 45798 . . . . . . . 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 45796 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦)))
8382ex 413 . . . . . . . . . . 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 1110 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
86 fcoi1 6685 . . . . . . . . 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 2777 . . . . . . 7 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)
89883exp 1118 . . . . . 6 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ (Base‘𝑥))) = 𝑔)))
90893ad2ant2 1133 . . . . 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 415 . . . 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 1110 . . 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 408 . 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 1198 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑦𝐵)
955, 1, 33, 21, 35, 94rngchomALTV 45795 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHomo 𝑦))
9695eleq2d 2823 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHomo 𝑦)))
9796biimpd 228 . . . . . . . . . 10 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHomo 𝑦)))
98973exp 1118 . . . . . . . . 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 1133 . . . . . . 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 1110 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉𝑔 ∈ (𝑥 RngHomo 𝑦)))
103102impcom 408 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥 RngHomo 𝑦))
104 rnghmco 45717 . . . 4 ((𝑔 ∈ (𝑥 RngHomo 𝑦) ∧ 𝑓 ∈ (𝑤 RngHomo 𝑥)) → (𝑔𝑓) ∈ (𝑤 RngHomo 𝑦))
105103, 44, 104syl2anc 584 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔𝑓) ∈ (𝑤 RngHomo 𝑦))
106 simp2l 1198 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
107106adantl 482 . . . 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 45798 . . 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 45795 . . 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 2853 . 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 6190 . . . 4 ((𝑔) ∘ 𝑓) = ( ∘ (𝑔𝑓))
112 simp2r 1199 . . . . . 6 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
113112adantl 482 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧𝐵)
114 simp2r 1199 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → 𝑧𝐵)
1155, 1, 33, 21, 94, 114rngchomALTV 45795 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RngHomo 𝑧))
116115eleq2d 2823 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ ∈ (𝑦 RngHomo 𝑧)))
117116biimpd 228 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ∈ (𝑦 RngHomo 𝑧)))
1181173exp 1118 . . . . . . . . . . 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 1134 . . . . . . . . 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 1110 . . . . . . 7 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 ∈ (𝑦 RngHomo 𝑧)))
123122impcom 408 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ∈ (𝑦 RngHomo 𝑧))
124 rnghmco 45717 . . . . . 6 (( ∈ (𝑦 RngHomo 𝑧) ∧ 𝑔 ∈ (𝑥 RngHomo 𝑦)) → (𝑔) ∈ (𝑥 RngHomo 𝑧))
125123, 103, 124syl2anc 584 . . . . 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 45798 . . . 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 45798 . . . 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 2803 . . 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 45798 . . . 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 7330 . . 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 7331 . . 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 2787 . 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 17460 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1540  wcel 2105  Vcvv 3441  cin 3896  cop 4577  cmpt 5170   I cid 5506  cres 5609  ccom 5611  wf 6461  cfv 6465  (class class class)co 7315  Basecbs 16982  Hom chom 17043  compcco 17044  Catccat 17443  Idccid 17444  Rngcrng 45684   RngHomo crngh 45695  RngCatALTVcrngcALTV 45768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628  ax-cnex 11000  ax-resscn 11001  ax-1cn 11002  ax-icn 11003  ax-addcl 11004  ax-addrcl 11005  ax-mulcl 11006  ax-mulrcl 11007  ax-mulcom 11008  ax-addass 11009  ax-mulass 11010  ax-distr 11011  ax-i2m1 11012  ax-1ne0 11013  ax-1rid 11014  ax-rnegex 11015  ax-rrecex 11016  ax-cnre 11017  ax-pre-lttri 11018  ax-pre-lttrn 11019  ax-pre-ltadd 11020  ax-pre-mulgt0 11021
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7272  df-ov 7318  df-oprab 7319  df-mpo 7320  df-om 7758  df-1st 7876  df-2nd 7877  df-frecs 8144  df-wrecs 8175  df-recs 8249  df-rdg 8288  df-1o 8344  df-er 8546  df-map 8665  df-en 8782  df-dom 8783  df-sdom 8784  df-fin 8785  df-pnf 11084  df-mnf 11085  df-xr 11086  df-ltxr 11087  df-le 11088  df-sub 11280  df-neg 11281  df-nn 12047  df-2 12109  df-3 12110  df-4 12111  df-5 12112  df-6 12113  df-7 12114  df-8 12115  df-9 12116  df-n0 12307  df-z 12393  df-dec 12511  df-uz 12656  df-fz 13313  df-struct 16918  df-sets 16935  df-slot 16953  df-ndx 16965  df-base 16983  df-plusg 17045  df-hom 17056  df-cco 17057  df-0g 17222  df-cat 17447  df-cid 17448  df-mgm 18396  df-sgrp 18445  df-mnd 18456  df-mhm 18500  df-grp 18649  df-ghm 18901  df-abl 19457  df-mgp 19789  df-mgmhm 45585  df-rng0 45685  df-rnghomo 45697  df-rngcALTV 45770
This theorem is referenced by:  rngccatALTV  45800  rngcidALTV  45801  rhmsubcALTVlem3  45916
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