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Theorem setccatid 17339
Description: Lemma for setccat 17340. (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 17333 . 2 (𝑈𝑉𝑈 = (Base‘𝐶))
4 eqidd 2802 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
5 eqidd 2802 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
61fvexi 6663 . . 3 𝐶 ∈ V
76a1i 11 . 2 (𝑈𝑉𝐶 ∈ V)
8 biid 264 . 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 6631 . . . 4 ( I ↾ 𝑥):𝑥1-1-onto𝑥
10 f1of 6594 . . . 4 (( I ↾ 𝑥):𝑥1-1-onto𝑥 → ( I ↾ 𝑥):𝑥𝑥)
119, 10mp1i 13 . . 3 ((𝑈𝑉𝑥𝑈) → ( I ↾ 𝑥):𝑥𝑥)
12 simpl 486 . . . 4 ((𝑈𝑉𝑥𝑈) → 𝑈𝑉)
13 eqid 2801 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
14 simpr 488 . . . 4 ((𝑈𝑉𝑥𝑈) → 𝑥𝑈)
151, 12, 13, 14, 14elsetchom 17336 . . 3 ((𝑈𝑉𝑥𝑈) → (( I ↾ 𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥) ↔ ( I ↾ 𝑥):𝑥𝑥))
1611, 15mpbird 260 . 2 ((𝑈𝑉𝑥𝑈) → ( I ↾ 𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
17 simpl 486 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
18 eqid 2801 . . . 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 17336 . . . . 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 235 . . . 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 17338 . . 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 6531 . . . 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 2836 . 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 17336 . . . . 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 235 . . . 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 17338 . . 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 6530 . . . 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 2836 . 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 17338 . . 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 6509 . . . . 5 ((𝑔:𝑥𝑦𝑓:𝑤𝑥) → (𝑔𝑓):𝑤𝑦)
3932, 23, 38syl2anc 587 . . . 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 17336 . . . 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 260 . . 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 2893 . 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 6089 . . . 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 17336 . . . . . . 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 235 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → :𝑦𝑧)
48 fco 6509 . . . . . 6 ((:𝑦𝑧𝑔:𝑥𝑦) → (𝑔):𝑥𝑧)
4947, 32, 48syl2anc 587 . . . . 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 17338 . . . 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 17338 . . . 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 2862 . . 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 17338 . . . 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 7154 . . 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 7155 . . 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 2846 . 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 16947 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  Vcvv 3444  cop 4534  cmpt 5113   I cid 5427  cres 5525  ccom 5527  wf 6324  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  Hom chom 16571  compcco 16572  Catccat 16930  Idccid 16931  SetCatcsetc 17330
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-hom 16584  df-cco 16585  df-cat 16934  df-cid 16935  df-setc 17331
This theorem is referenced by:  setccat  17340  setcid  17341
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