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Theorem catccatid 16959
Description: Lemma for catccat 16961. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
catccatid.c 𝐶 = (CatCat‘𝑈)
catccatid.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
catccatid (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ (idfunc𝑥))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝑈   𝑥,𝑉

Proof of Theorem catccatid
Dummy variables 𝑓 𝑔 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catccatid.b . . 3 𝐵 = (Base‘𝐶)
21a1i 11 . 2 (𝑈𝑉𝐵 = (Base‘𝐶))
3 eqidd 2772 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
4 eqidd 2772 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
5 catccatid.c . . . 4 𝐶 = (CatCat‘𝑈)
65fvexi 6345 . . 3 𝐶 ∈ V
76a1i 11 . 2 (𝑈𝑉𝐶 ∈ V)
8 biid 251 . 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 id 22 . . . . . . 7 (𝑈𝑉𝑈𝑉)
105, 1, 9catcbas 16954 . . . . . 6 (𝑈𝑉𝐵 = (𝑈 ∩ Cat))
11 inss2 3982 . . . . . 6 (𝑈 ∩ Cat) ⊆ Cat
1210, 11syl6eqss 3804 . . . . 5 (𝑈𝑉𝐵 ⊆ Cat)
1312sselda 3752 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥 ∈ Cat)
14 eqid 2771 . . . . 5 (idfunc𝑥) = (idfunc𝑥)
1514idfucl 16748 . . . 4 (𝑥 ∈ Cat → (idfunc𝑥) ∈ (𝑥 Func 𝑥))
1613, 15syl 17 . . 3 ((𝑈𝑉𝑥𝐵) → (idfunc𝑥) ∈ (𝑥 Func 𝑥))
17 simpl 468 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑈𝑉)
18 eqid 2771 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
19 simpr 471 . . . 4 ((𝑈𝑉𝑥𝐵) → 𝑥𝐵)
205, 1, 17, 18, 19, 19catchom 16956 . . 3 ((𝑈𝑉𝑥𝐵) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥 Func 𝑥))
2116, 20eleqtrrd 2853 . 2 ((𝑈𝑉𝑥𝐵) → (idfunc𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
22 simpl 468 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
23 eqid 2771 . . . 4 (comp‘𝐶) = (comp‘𝐶)
24 simpr1l 1290 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤𝐵)
25 simpr1r 1292 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥𝐵)
26 simpr31 1344 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥))
275, 1, 22, 18, 24, 25catchom 16956 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑥) = (𝑤 Func 𝑥))
2826, 27eleqtrd 2852 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤 Func 𝑥))
2925, 16syldan 579 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (idfunc𝑥) ∈ (𝑥 Func 𝑥))
305, 1, 22, 23, 24, 25, 25, 28, 29catcco 16958 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((idfunc𝑥)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = ((idfunc𝑥) ∘func 𝑓))
3128, 14cofulid 16757 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((idfunc𝑥) ∘func 𝑓) = 𝑓)
3230, 31eqtrd 2805 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((idfunc𝑥)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
33 simpr2l 1294 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦𝐵)
34 simpr32 1346 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
355, 1, 22, 18, 25, 33catchom 16956 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 Func 𝑦))
3634, 35eleqtrd 2852 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥 Func 𝑦))
375, 1, 22, 23, 25, 25, 33, 29, 36catcco 16958 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)(idfunc𝑥)) = (𝑔func (idfunc𝑥)))
3836, 14cofurid 16758 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔func (idfunc𝑥)) = 𝑔)
3937, 38eqtrd 2805 . 2 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)(idfunc𝑥)) = 𝑔)
4028, 36cofucl 16755 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔func 𝑓) ∈ (𝑤 Func 𝑦))
415, 1, 22, 23, 24, 25, 33, 28, 36catcco 16958 . . 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) = (𝑔func 𝑓))
425, 1, 22, 18, 24, 33catchom 16956 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑤(Hom ‘𝐶)𝑦) = (𝑤 Func 𝑦))
4340, 41, 423eltr4d 2865 . 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))
44 simpr33 1348 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ∈ (𝑦(Hom ‘𝐶)𝑧))
45 simpr2r 1296 . . . . . . 7 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧𝐵)
465, 1, 22, 18, 33, 45catchom 16956 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦 Func 𝑧))
4744, 46eleqtrd 2852 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ∈ (𝑦 Func 𝑧))
4828, 36, 47cofuass 16756 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((func 𝑔) ∘func 𝑓) = (func (𝑔func 𝑓)))
4936, 47cofucl 16755 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (func 𝑔) ∈ (𝑥 Func 𝑧))
505, 1, 22, 23, 24, 25, 45, 28, 49catcco 16958 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((func 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((func 𝑔) ∘func 𝑓))
515, 1, 22, 23, 24, 33, 45, 40, 47catcco 16958 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔func 𝑓)) = (func (𝑔func 𝑓)))
5248, 50, 513eqtr4d 2815 . . 3 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((func 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔func 𝑓)))
535, 1, 22, 23, 25, 33, 45, 36, 47catcco 16958 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝐵𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) = (func 𝑔))
5453oveq1d 6811 . . 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) = ((func 𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓))
5541oveq2d 6812 . . 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)(𝑔func 𝑓)))
5652, 54, 553eqtr4d 2815 . 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)))
572, 3, 4, 7, 8, 21, 32, 39, 43, 56iscatd2 16549 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ (idfunc𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  cin 3722  cop 4323  cmpt 4864  cfv 6030  (class class class)co 6796  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532  Idccid 16533   Func cfunc 16721  idfunccidfu 16722  func ccofu 16723  CatCatccatc 16951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-7 11290  df-8 11291  df-9 11292  df-n0 11500  df-z 11585  df-dec 11701  df-uz 11894  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-hom 16174  df-cco 16175  df-cat 16536  df-cid 16537  df-func 16725  df-idfu 16726  df-cofu 16727  df-catc 16952
This theorem is referenced by:  catcid  16960  catccat  16961
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