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Theorem estrccatid 18176
Description: Lemma for estrccat 18177. (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 18169 . 2 (𝑈𝑉𝑈 = (Base‘𝐶))
4 eqidd 2738 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
5 eqidd 2738 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
61fvexi 6920 . . 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 6886 . . . 4 ( I ↾ (Base‘𝑥)):(Base‘𝑥)–1-1-onto→(Base‘𝑥)
10 f1of 6848 . . . 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 2737 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
14 simpr 484 . . . 4 ((𝑈𝑉𝑥𝑈) → 𝑥𝑈)
15 eqid 2737 . . . 4 (Base‘𝑥) = (Base‘𝑥)
161, 12, 13, 14, 14, 15, 15elestrchom 18172 . . 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 2737 . . . 4 (comp‘𝐶) = (comp‘𝐶)
20 simpr1l 1231 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤𝑈)
21 simpr1r 1232 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥𝑈)
22 eqid 2737 . . . 4 (Base‘𝑤) = (Base‘𝑤)
23 simpr31 1264 . . . . 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 18172 . . . . 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 232 . . . 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 18174 . . 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 6783 . . . 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 2777 . 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 1233 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦𝑈)
32 eqid 2737 . . . 4 (Base‘𝑦) = (Base‘𝑦)
33 simpr32 1265 . . . . 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 18172 . . . . 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 232 . . . 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 18174 . . 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 6782 . . . 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 2777 . 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 18174 . . 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 6760 . . . . 5 ((𝑔:(Base‘𝑥)⟶(Base‘𝑦) ∧ 𝑓:(Base‘𝑤)⟶(Base‘𝑥)) → (𝑔𝑓):(Base‘𝑤)⟶(Base‘𝑦))
4235, 25, 41syl2anc 584 . . . 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 18172 . . . 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 2841 . 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 6285 . . . 4 ((𝑔) ∘ 𝑓) = ( ∘ (𝑔𝑓))
47 simpr2r 1234 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧𝑈)
48 eqid 2737 . . . . 5 (Base‘𝑧) = (Base‘𝑧)
49 simpr33 1266 . . . . . . 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 18172 . . . . . . 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 232 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → :(Base‘𝑦)⟶(Base‘𝑧))
52 fco 6760 . . . . . 6 ((:(Base‘𝑦)⟶(Base‘𝑧) ∧ 𝑔:(Base‘𝑥)⟶(Base‘𝑦)) → (𝑔):(Base‘𝑥)⟶(Base‘𝑧))
5351, 35, 52syl2anc 584 . . . . 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 18174 . . . 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 18174 . . . 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 2803 . . 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 18174 . . . 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 7446 . . 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 7447 . . 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 2787 . 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 17724 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ (Base‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cmpt 5225   I cid 5577  cres 5687  ccom 5689  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708  ExtStrCatcestrc 18166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-cat 17711  df-cid 17712  df-estrc 18167
This theorem is referenced by:  estrccat  18177  estrcid  18178
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