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Theorem setccatid 18081
Description: Lemma for setccat 18082. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypothesis
Ref Expression
setccat.c 𝐶 = (SetCat‘𝑈)
Assertion
Ref Expression
setccatid (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ 𝑥))))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑈   𝑥,𝑉

Proof of Theorem setccatid
Dummy variables 𝑓 𝑔 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 setccat.c . . 3 𝐶 = (SetCat‘𝑈)
2 id 22 . . 3 (𝑈𝑉𝑈𝑉)
31, 2setcbas 18075 . 2 (𝑈𝑉𝑈 = (Base‘𝐶))
4 eqidd 2726 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
5 eqidd 2726 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
61fvexi 6910 . . 3 𝐶 ∈ V
76a1i 11 . 2 (𝑈𝑉𝐶 ∈ V)
8 biid 260 . 2 (((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))) ↔ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧))))
9 f1oi 6876 . . . 4 ( I ↾ 𝑥):𝑥1-1-onto𝑥
10 f1of 6838 . . . 4 (( I ↾ 𝑥):𝑥1-1-onto𝑥 → ( I ↾ 𝑥):𝑥𝑥)
119, 10mp1i 13 . . 3 ((𝑈𝑉𝑥𝑈) → ( I ↾ 𝑥):𝑥𝑥)
12 simpl 481 . . . 4 ((𝑈𝑉𝑥𝑈) → 𝑈𝑉)
13 eqid 2725 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
14 simpr 483 . . . 4 ((𝑈𝑉𝑥𝑈) → 𝑥𝑈)
151, 12, 13, 14, 14elsetchom 18078 . . 3 ((𝑈𝑉𝑥𝑈) → (( I ↾ 𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥) ↔ ( I ↾ 𝑥):𝑥𝑥))
1611, 15mpbird 256 . 2 ((𝑈𝑉𝑥𝑈) → ( I ↾ 𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
17 simpl 481 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
18 eqid 2725 . . . 4 (comp‘𝐶) = (comp‘𝐶)
19 simpr1l 1227 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤𝑈)
20 simpr1r 1228 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥𝑈)
21 simpr31 1260 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥))
221, 17, 13, 19, 20elsetchom 18078 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓:𝑤𝑥))
2321, 22mpbid 231 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓:𝑤𝑥)
249, 10mp1i 13 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( I ↾ 𝑥):𝑥𝑥)
251, 17, 18, 19, 20, 20, 23, 24setcco 18080 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ 𝑥)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = (( I ↾ 𝑥) ∘ 𝑓))
26 fcoi2 6772 . . . 4 (𝑓:𝑤𝑥 → (( I ↾ 𝑥) ∘ 𝑓) = 𝑓)
2723, 26syl 17 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ 𝑥) ∘ 𝑓) = 𝑓)
2825, 27eqtrd 2765 . 2 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ 𝑥)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
29 simpr2l 1229 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦𝑈)
30 simpr32 1261 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
311, 17, 13, 20, 29elsetchom 18078 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔:𝑥𝑦))
3230, 31mpbid 231 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔:𝑥𝑦)
331, 17, 18, 20, 20, 29, 24, 32setcco 18080 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ 𝑥)) = (𝑔 ∘ ( I ↾ 𝑥)))
34 fcoi1 6771 . . . 4 (𝑔:𝑥𝑦 → (𝑔 ∘ ( I ↾ 𝑥)) = 𝑔)
3532, 34syl 17 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ ( I ↾ 𝑥)) = 𝑔)
3633, 35eqtrd 2765 . 2 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( I ↾ 𝑥)) = 𝑔)
371, 17, 18, 19, 20, 29, 23, 32setcco 18080 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) = (𝑔𝑓))
38 fco 6747 . . . . 5 ((𝑔:𝑥𝑦𝑓:𝑤𝑥) → (𝑔𝑓):𝑤𝑦)
3932, 23, 38syl2anc 582 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔𝑓):𝑤𝑦)
401, 17, 13, 19, 29elsetchom 18078 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑔𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦) ↔ (𝑔𝑓):𝑤𝑦))
4139, 40mpbird 256 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦))
4237, 41eqeltrd 2825 . 2 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (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))
43 coass 6271 . . . 4 ((𝑔) ∘ 𝑓) = ( ∘ (𝑔𝑓))
44 simpr2r 1230 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧𝑈)
45 simpr33 1262 . . . . . . 7 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ∈ (𝑦(Hom ‘𝐶)𝑧))
461, 17, 13, 29, 44elsetchom 18078 . . . . . . 7 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ :𝑦𝑧))
4745, 46mpbid 231 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → :𝑦𝑧)
48 fco 6747 . . . . . 6 ((:𝑦𝑧𝑔:𝑥𝑦) → (𝑔):𝑥𝑧)
4947, 32, 48syl2anc 582 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔):𝑥𝑧)
501, 17, 18, 19, 20, 44, 23, 49setcco 18080 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((𝑔) ∘ 𝑓))
511, 17, 18, 19, 29, 44, 39, 47setcco 18080 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔𝑓)) = ( ∘ (𝑔𝑓)))
5243, 50, 513eqtr4a 2791 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (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)))
531, 17, 18, 20, 29, 44, 32, 47setcco 18080 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) = (𝑔))
5453oveq1d 7434 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (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))
5537oveq2d 7435 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (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)))
5652, 54, 553eqtr4d 2775 . 2 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (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)))
573, 4, 5, 7, 8, 16, 28, 36, 42, 56iscatd2 17669 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3461  cop 4636  cmpt 5232   I cid 5575  cres 5680  ccom 5682  wf 6545  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  Hom chom 17252  compcco 17253  Catccat 17652  Idccid 17653  SetCatcsetc 18072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17189  df-hom 17265  df-cco 17266  df-cat 17656  df-cid 17657  df-setc 18073
This theorem is referenced by:  setccat  18082  setcid  18083
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