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Theorem rngccatidALTV 48115
Description: Lemma for rngccatALTV 48116. (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 2735 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4 eqidd 2735 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
5 rngccatALTV.c . . . 4 𝐶 = (RngCatALTV‘𝑈)
65fvexi 6920 . . 3 𝐶 ∈ V
76a1i 11 . 2 (𝑈𝑉𝐶 ∈ V)
8 biid 261 . 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 482 . . . . . 6 ((𝑈𝑉𝑥𝐵) → 𝑈𝑉)
105, 1, 9rngcbasALTV 48109 . . . . 5 ((𝑈𝑉𝑥𝐵) → 𝐵 = (𝑈 ∩ Rng))
11 eleq2 2827 . . . . . . . 8 (𝐵 = (𝑈 ∩ Rng) → (𝑥𝐵𝑥 ∈ (𝑈 ∩ Rng)))
12 elin 3978 . . . . . . . . 9 (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥𝑈𝑥 ∈ Rng))
1312simprbi 496 . . . . . . . 8 (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥 ∈ Rng)
1411, 13biimtrdi 253 . . . . . . 7 (𝐵 = (𝑈 ∩ Rng) → (𝑥𝐵𝑥 ∈ Rng))
1514com12 32 . . . . . 6 (𝑥𝐵 → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
1615adantl 481 . . . . 5 ((𝑈𝑉𝑥𝐵) → (𝐵 = (𝑈 ∩ Rng) → 𝑥 ∈ Rng))
1710, 16mpd 15 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥 ∈ Rng)
18 eqid 2734 . . . . 5 (Base‘𝑥) = (Base‘𝑥)
1918idrnghm 20474 . . . 4 (𝑥 ∈ Rng → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 𝑥))
2017, 19syl 17 . . 3 ((𝑈𝑉𝑥𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 𝑥))
21 eqid 2734 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
22 simpr 484 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥𝐵)
235, 1, 9, 21, 22, 22rngchomALTV 48111 . . 3 ((𝑈𝑉𝑥𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 RngHom 𝑥))
2420, 23eleqtrrd 2841 . 2 ((𝑈𝑉𝑥𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
25 simpl 482 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
26 eqid 2734 . . . 4 (comp‘𝐶) = (comp‘𝐶)
27 simpl 482 . . . . . 6 ((𝑤𝐵𝑥𝐵) → 𝑤𝐵)
28273ad2ant1 1132 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑤𝐵)
2928adantl 481 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤𝐵)
30 simpr 484 . . . . . 6 ((𝑤𝐵𝑥𝐵) → 𝑥𝐵)
31303ad2ant1 1132 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝐵)
3231adantl 481 . . . 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 48111 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 RngHom 𝑥))
3736eleq2d 2824 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓 ∈ (𝑤 RngHom 𝑥)))
3837biimpd 229 . . . . . . . . . 10 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RngHom 𝑥)))
39383exp 1118 . . . . . . . . 9 (𝑈𝑉 → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓 ∈ (𝑤 RngHom 𝑥)))))
4039com14 96 . . . . . . . 8 (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑓 ∈ (𝑤 RngHom 𝑥)))))
41403ad2ant1 1132 . . . . . . 7 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑓 ∈ (𝑤 RngHom 𝑥)))))
4241com13 88 . . . . . 6 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉𝑓 ∈ (𝑤 RngHom 𝑥)))))
43423imp 1110 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉𝑓 ∈ (𝑤 RngHom 𝑥)))
4443impcom 407 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤 RngHom 𝑥))
4520expcom 413 . . . . . . 7 (𝑥𝐵 → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 𝑥)))
4645adantl 481 . . . . . 6 ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 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 RngHom 𝑥)))
4847impcom 407 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 𝑥))
495, 1, 25, 26, 29, 32, 32, 44, 48rngccoALTV 48114 . . 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 482 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑈𝑉)
51 simprl 771 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑤𝐵)
52 simprr 773 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → 𝑥𝐵)
535, 1, 50, 21, 51, 52elrngchomALTV 48112 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑤𝐵𝑥𝐵)) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥)))
5453ex 412 . . . . . . . . . 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 6783 . . . . . . . . 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 407 . . 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 2774 . 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 480 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑥𝐵)
66653ad2ant2 1133 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑥𝐵)
67 simprl 771 . . . . . . . . . 10 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
68673ad2ant2 1133 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑦𝐵)
6946adantr 480 . . . . . . . . . . 11 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 𝑥)))
7069a1i 11 . . . . . . . . . 10 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉 → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 𝑥))))
71703imp 1110 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 𝑥))
72 simpl 482 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑈𝑉)
7365adantl 481 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑥𝐵)
7467adantl 481 . . . . . . . . . . . . . . 15 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → 𝑦𝐵)
755, 1, 72, 21, 73, 74rngchomALTV 48111 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHom 𝑦))
7675eleq2d 2824 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHom 𝑦)))
7776biimpd 229 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHom 𝑦)))
7877ex 412 . . . . . . . . . . 11 (𝑈𝑉 → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHom 𝑦))))
7978com13 88 . . . . . . . . . 10 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → (𝑈𝑉𝑔 ∈ (𝑥 RngHom 𝑦))))
80793imp 1110 . . . . . . . . 9 ((𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑈𝑉) → 𝑔 ∈ (𝑥 RngHom 𝑦))
815, 1, 64, 26, 66, 66, 68, 71, 80rngccoALTV 48114 . . . . . . . 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 48112 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦)))
8382ex 412 . . . . . . . . . . 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 6782 . . . . . . . . 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 2774 . . . . . . 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 414 . . . 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 407 . 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 48111 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 RngHom 𝑦))
9695eleq2d 2824 . . . . . . . . . . 11 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔 ∈ (𝑥 RngHom 𝑦)))
9796biimpd 229 . . . . . . . . . 10 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHom 𝑦)))
98973exp 1118 . . . . . . . . 9 (𝑈𝑉 → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → 𝑔 ∈ (𝑥 RngHom 𝑦)))))
9998com14 96 . . . . . . . 8 (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑔 ∈ (𝑥 RngHom 𝑦)))))
100993ad2ant2 1133 . . . . . . 7 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉𝑔 ∈ (𝑥 RngHom 𝑦)))))
101100com13 88 . . . . . 6 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉𝑔 ∈ (𝑥 RngHom 𝑦)))))
1021013imp 1110 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉𝑔 ∈ (𝑥 RngHom 𝑦)))
103102impcom 407 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥 RngHom 𝑦))
104 rnghmco 20473 . . . 4 ((𝑔 ∈ (𝑥 RngHom 𝑦) ∧ 𝑓 ∈ (𝑤 RngHom 𝑥)) → (𝑔𝑓) ∈ (𝑤 RngHom 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 RngHom 𝑦))
106 simp2l 1198 . . . . 5 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
107106adantl 481 . . . 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 48114 . . 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 48111 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑦) = (𝑤 RngHom 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 6286 . . . 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 481 . . . . 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 48111 . . . . . . . . . . . . . 14 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 RngHom 𝑧))
116115eleq2d 2824 . . . . . . . . . . . . 13 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ ∈ (𝑦 RngHom 𝑧)))
117116biimpd 229 . . . . . . . . . . . 12 ((𝑈𝑉 ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑤𝐵𝑥𝐵)) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ∈ (𝑦 RngHom 𝑧)))
1181173exp 1118 . . . . . . . . . . 11 (𝑈𝑉 → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ∈ (𝑦 RngHom 𝑧)))))
119118com14 96 . . . . . . . . . 10 ( ∈ (𝑦(Hom ‘𝐶)𝑧) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 ∈ (𝑦 RngHom 𝑧)))))
1201193ad2ant3 1134 . . . . . . . . 9 ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑦𝐵𝑧𝐵) → ((𝑤𝐵𝑥𝐵) → (𝑈𝑉 ∈ (𝑦 RngHom 𝑧)))))
121120com13 88 . . . . . . . 8 ((𝑤𝐵𝑥𝐵) → ((𝑦𝐵𝑧𝐵) → ((𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑈𝑉 ∈ (𝑦 RngHom 𝑧)))))
1221213imp 1110 . . . . . . 7 (((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑈𝑉 ∈ (𝑦 RngHom 𝑧)))
123122impcom 407 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ∈ (𝑦 RngHom 𝑧))
124 rnghmco 20473 . . . . . 6 (( ∈ (𝑦 RngHom 𝑧) ∧ 𝑔 ∈ (𝑥 RngHom 𝑦)) → (𝑔) ∈ (𝑥 RngHom 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 RngHom 𝑧))
1265, 1, 25, 26, 29, 32, 113, 44, 125rngccoALTV 48114 . . . 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 48114 . . . 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 2800 . . 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 48114 . . . 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 7445 . . 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 7446 . . 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 2784 . 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 17725 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961  cop 4636  cmpt 5230   I cid 5581  cres 5690  ccom 5692  wf 6558  cfv 6562  (class class class)co 7430  Basecbs 17244  Hom chom 17308  compcco 17309  Catccat 17708  Idccid 17709  Rngcrng 20169   RngHom crnghm 20450  RngCatALTVcrngcALTV 48106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-plusg 17310  df-hom 17321  df-cco 17322  df-0g 17487  df-cat 17712  df-cid 17713  df-mgm 18665  df-mgmhm 18717  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-grp 18966  df-ghm 19243  df-abl 19815  df-mgp 20152  df-rng 20170  df-rnghm 20452  df-rngcALTV 48107
This theorem is referenced by:  rngccatALTV  48116  rngcidALTV  48117  rhmsubcALTVlem3  48126
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