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Theorem estrccatid 18085
Description: Lemma for estrccat 18086. (Contributed by AV, 8-Mar-2020.)
Hypothesis
Ref Expression
estrccat.c 𝐶 = (ExtStrCat‘𝑈)
Assertion
Ref Expression
estrccatid (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ (Base‘𝑥)))))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑈   𝑥,𝑉

Proof of Theorem estrccatid
Dummy variables 𝑓 𝑔 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 estrccat.c . . 3 𝐶 = (ExtStrCat‘𝑈)
2 id 22 . . 3 (𝑈𝑉𝑈𝑉)
31, 2estrcbas 18078 . 2 (𝑈𝑉𝑈 = (Base‘𝐶))
4 eqidd 2725 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
5 eqidd 2725 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
61fvexi 6895 . . 3 𝐶 ∈ V
76a1i 11 . 2 (𝑈𝑉𝐶 ∈ V)
8 biid 261 . 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 6861 . . . 4 ( I ↾ (Base‘𝑥)):(Base‘𝑥)–1-1-onto→(Base‘𝑥)
10 f1of 6823 . . . 4 (( I ↾ (Base‘𝑥)):(Base‘𝑥)–1-1-onto→(Base‘𝑥) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥))
119, 10mp1i 13 . . 3 ((𝑈𝑉𝑥𝑈) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥))
12 simpl 482 . . . 4 ((𝑈𝑉𝑥𝑈) → 𝑈𝑉)
13 eqid 2724 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
14 simpr 484 . . . 4 ((𝑈𝑉𝑥𝑈) → 𝑥𝑈)
15 eqid 2724 . . . 4 (Base‘𝑥) = (Base‘𝑥)
161, 12, 13, 14, 14, 15, 15elestrchom 18081 . . 3 ((𝑈𝑉𝑥𝑈) → (( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥) ↔ ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥)))
1711, 16mpbird 257 . 2 ((𝑈𝑉𝑥𝑈) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
18 simpl 482 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
19 eqid 2724 . . . 4 (comp‘𝐶) = (comp‘𝐶)
20 simpr1l 1227 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤𝑈)
21 simpr1r 1228 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥𝑈)
22 eqid 2724 . . . 4 (Base‘𝑤) = (Base‘𝑤)
23 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))
241, 18, 13, 20, 21, 22, 15elestrchom 18081 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ↔ 𝑓:(Base‘𝑤)⟶(Base‘𝑥)))
2523, 24mpbid 231 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥))
269, 10mp1i 13 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥))
271, 18, 19, 20, 21, 21, 22, 15, 15, 25, 26estrcco 18083 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (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))
28 fcoi2 6756 . . . 4 (𝑓:(Base‘𝑤)⟶(Base‘𝑥) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
2925, 28syl 17 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
3027, 29eqtrd 2764 . 2 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥))(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
31 simpr2l 1229 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦𝑈)
32 eqid 2724 . . . 4 (Base‘𝑦) = (Base‘𝑦)
33 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))
341, 18, 13, 21, 31, 15, 32elestrchom 18081 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ 𝑔:(Base‘𝑥)⟶(Base‘𝑦)))
3533, 34mpbid 231 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
361, 18, 19, 21, 21, 31, 15, 15, 32, 26, 35estrcco 18083 . . 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 ↾ (Base‘𝑥))) = (𝑔 ∘ ( I ↾ (Base‘𝑥))))
37 fcoi1 6755 . . . 4 (𝑔:(Base‘𝑥)⟶(Base‘𝑦) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
3835, 37syl 17 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
3936, 38eqtrd 2764 . 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 ↾ (Base‘𝑥))) = 𝑔)
401, 18, 19, 20, 21, 31, 22, 15, 32, 25, 35estrcco 18083 . . 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))
41 fco 6731 . . . . 5 ((𝑔:(Base‘𝑥)⟶(Base‘𝑦) ∧ 𝑓:(Base‘𝑤)⟶(Base‘𝑥)) → (𝑔𝑓):(Base‘𝑤)⟶(Base‘𝑦))
4235, 25, 41syl2anc 583 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔𝑓):(Base‘𝑤)⟶(Base‘𝑦))
431, 18, 13, 20, 31, 22, 32elestrchom 18081 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ((𝑔𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦) ↔ (𝑔𝑓):(Base‘𝑤)⟶(Base‘𝑦)))
4442, 43mpbird 257 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦))
4540, 44eqeltrd 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))
46 coass 6254 . . . 4 ((𝑔) ∘ 𝑓) = ( ∘ (𝑔𝑓))
47 simpr2r 1230 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧𝑈)
48 eqid 2724 . . . . 5 (Base‘𝑧) = (Base‘𝑧)
49 simpr33 1262 . . . . . . 7 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ∈ (𝑦(Hom ‘𝐶)𝑧))
501, 18, 13, 31, 47, 32, 48elestrchom 18081 . . . . . . 7 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ( ∈ (𝑦(Hom ‘𝐶)𝑧) ↔ :(Base‘𝑦)⟶(Base‘𝑧)))
5149, 50mpbid 231 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → :(Base‘𝑦)⟶(Base‘𝑧))
52 fco 6731 . . . . . 6 ((:(Base‘𝑦)⟶(Base‘𝑧) ∧ 𝑔:(Base‘𝑥)⟶(Base‘𝑦)) → (𝑔):(Base‘𝑥)⟶(Base‘𝑧))
5351, 35, 52syl2anc 583 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔):(Base‘𝑥)⟶(Base‘𝑧))
541, 18, 19, 20, 21, 47, 22, 15, 48, 25, 53estrcco 18083 . . . 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))
551, 18, 19, 20, 31, 47, 22, 32, 48, 42, 51estrcco 18083 . . . 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)))
5646, 54, 553eqtr4a 2790 . . 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)))
571, 18, 19, 21, 31, 47, 15, 32, 48, 35, 51estrcco 18083 . . . 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))
5857oveq1d 7416 . . 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))
5940oveq2d 7417 . . 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)))
6056, 58, 593eqtr4d 2774 . 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)))
613, 4, 5, 7, 8, 17, 30, 39, 45, 60iscatd2 17624 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ (Base‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  cop 4626  cmpt 5221   I cid 5563  cres 5668  ccom 5670  wf 6529  1-1-ontowf1o 6532  cfv 6533  (class class class)co 7401  Basecbs 17143  Hom chom 17207  compcco 17208  Catccat 17607  Idccid 17608  ExtStrCatcestrc 18075
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 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17079  df-slot 17114  df-ndx 17126  df-base 17144  df-hom 17220  df-cco 17221  df-cat 17611  df-cid 17612  df-estrc 18076
This theorem is referenced by:  estrccat  18086  estrcid  18087
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