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Theorem estrccatid 16541
Description: Lemma for estrccat 16542. (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 16534 . 2 (𝑈𝑉𝑈 = (Base‘𝐶))
4 eqidd 2610 . 2 (𝑈𝑉 → (Hom ‘𝐶) = (Hom ‘𝐶))
5 eqidd 2610 . 2 (𝑈𝑉 → (comp‘𝐶) = (comp‘𝐶))
6 fvex 6098 . . . 4 (ExtStrCat‘𝑈) ∈ V
71, 6eqeltri 2683 . . 3 𝐶 ∈ V
87a1i 11 . 2 (𝑈𝑉𝐶 ∈ V)
9 biid 249 . 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))))
10 f1oi 6071 . . . 4 ( I ↾ (Base‘𝑥)):(Base‘𝑥)–1-1-onto→(Base‘𝑥)
11 f1of 6035 . . . 4 (( I ↾ (Base‘𝑥)):(Base‘𝑥)–1-1-onto→(Base‘𝑥) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥))
1210, 11mp1i 13 . . 3 ((𝑈𝑉𝑥𝑈) → ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥))
13 simpl 471 . . . 4 ((𝑈𝑉𝑥𝑈) → 𝑈𝑉)
14 eqid 2609 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
15 simpr 475 . . . 4 ((𝑈𝑉𝑥𝑈) → 𝑥𝑈)
16 eqid 2609 . . . 4 (Base‘𝑥) = (Base‘𝑥)
171, 13, 14, 15, 15, 16, 16elestrchom 16537 . . 3 ((𝑈𝑉𝑥𝑈) → (( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥) ↔ ( I ↾ (Base‘𝑥)):(Base‘𝑥)⟶(Base‘𝑥)))
1812, 17mpbird 245 . 2 ((𝑈𝑉𝑥𝑈) → ( I ↾ (Base‘𝑥)) ∈ (𝑥(Hom ‘𝐶)𝑥))
19 simpl 471 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑈𝑉)
20 eqid 2609 . . . 4 (comp‘𝐶) = (comp‘𝐶)
21 simpr1l 1110 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑤𝑈)
22 simpr1r 1111 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑥𝑈)
23 eqid 2609 . . . 4 (Base‘𝑤) = (Base‘𝑤)
24 simpr31 1143 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥))
251, 19, 14, 21, 22, 23, 16elestrchom 16537 . . . . 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)))
2624, 25mpbid 220 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑓:(Base‘𝑤)⟶(Base‘𝑥))
2710, 11mp1i 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))
281, 19, 20, 21, 22, 22, 23, 16, 16, 26, 27estrcco 16539 . . 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))
29 fcoi2 5977 . . . 4 (𝑓:(Base‘𝑤)⟶(Base‘𝑥) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
3026, 29syl 17 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (( I ↾ (Base‘𝑥)) ∘ 𝑓) = 𝑓)
3128, 30eqtrd 2643 . 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)
32 simpr2l 1112 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑦𝑈)
33 eqid 2609 . . . 4 (Base‘𝑦) = (Base‘𝑦)
34 simpr32 1144 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
351, 19, 14, 22, 32, 16, 33elestrchom 16537 . . . . 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)))
3634, 35mpbid 220 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑔:(Base‘𝑥)⟶(Base‘𝑦))
371, 19, 20, 22, 22, 32, 16, 16, 33, 27, 36estrcco 16539 . . 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))))
38 fcoi1 5976 . . . 4 (𝑔:(Base‘𝑥)⟶(Base‘𝑦) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
3936, 38syl 17 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔 ∘ ( I ↾ (Base‘𝑥))) = 𝑔)
4037, 39eqtrd 2643 . 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)
411, 19, 20, 21, 22, 32, 23, 16, 33, 26, 36estrcco 16539 . . 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))
42 fco 5957 . . . . 5 ((𝑔:(Base‘𝑥)⟶(Base‘𝑦) ∧ 𝑓:(Base‘𝑤)⟶(Base‘𝑥)) → (𝑔𝑓):(Base‘𝑤)⟶(Base‘𝑦))
4336, 26, 42syl2anc 690 . . . 4 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔𝑓):(Base‘𝑤)⟶(Base‘𝑦))
441, 19, 14, 21, 32, 23, 33elestrchom 16537 . . . 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)))
4543, 44mpbird 245 . . 3 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔𝑓) ∈ (𝑤(Hom ‘𝐶)𝑦))
4641, 45eqeltrd 2687 . 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))
47 coass 5557 . . . 4 ((𝑔) ∘ 𝑓) = ( ∘ (𝑔𝑓))
48 simpr2r 1113 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → 𝑧𝑈)
49 eqid 2609 . . . . 5 (Base‘𝑧) = (Base‘𝑧)
50 simpr33 1145 . . . . . . 7 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → ∈ (𝑦(Hom ‘𝐶)𝑧))
511, 19, 14, 32, 48, 33, 49elestrchom 16537 . . . . . . 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)))
5250, 51mpbid 220 . . . . . 6 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → :(Base‘𝑦)⟶(Base‘𝑧))
53 fco 5957 . . . . . 6 ((:(Base‘𝑦)⟶(Base‘𝑧) ∧ 𝑔:(Base‘𝑥)⟶(Base‘𝑦)) → (𝑔):(Base‘𝑥)⟶(Base‘𝑧))
5452, 36, 53syl2anc 690 . . . . 5 ((𝑈𝑉 ∧ ((𝑤𝑈𝑥𝑈) ∧ (𝑦𝑈𝑧𝑈) ∧ (𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ ∈ (𝑦(Hom ‘𝐶)𝑧)))) → (𝑔):(Base‘𝑥)⟶(Base‘𝑧))
551, 19, 20, 21, 22, 48, 23, 16, 49, 26, 54estrcco 16539 . . . 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))
561, 19, 20, 21, 32, 48, 23, 33, 49, 43, 52estrcco 16539 . . . 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)))
5747, 55, 563eqtr4a 2669 . . 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)))
581, 19, 20, 22, 32, 48, 16, 33, 49, 36, 52estrcco 16539 . . . 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))
5958oveq1d 6542 . . 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))
6041oveq2d 6543 . . 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)))
6157, 59, 603eqtr4d 2653 . 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)))
623, 4, 5, 8, 9, 18, 31, 40, 46, 61iscatd2 16111 1 (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ (Base‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  cop 4130  cmpt 4637   I cid 4938  cres 5030  ccom 5032  wf 5786  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  Basecbs 15641  Hom chom 15725  compcco 15726  Catccat 16094  Idccid 16095  ExtStrCatcestrc 16531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-fz 12153  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-hom 15739  df-cco 15740  df-cat 16098  df-cid 16099  df-estrc 16532
This theorem is referenced by:  estrccat  16542  estrcid  16543
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