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Theorem initoeu2 16877
Description: Initial objects are essentially unique, if A is an initial object, then so is every object that is isomorphic to A. Proposition 7.3 (2) in [Adamek] p. 102. (Contributed by AV, 10-Apr-2020.)
Hypotheses
Ref Expression
initoeu1.c (𝜑𝐶 ∈ Cat)
initoeu1.a (𝜑𝐴 ∈ (InitO‘𝐶))
initoeu2.i (𝜑𝐴( ≃𝑐𝐶)𝐵)
Assertion
Ref Expression
initoeu2 (𝜑𝐵 ∈ (InitO‘𝐶))

Proof of Theorem initoeu2
Dummy variables 𝑎 𝑔 𝑏 𝑓 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 initoeu2.i . 2 (𝜑𝐴( ≃𝑐𝐶)𝐵)
2 initoeu1.c . . . 4 (𝜑𝐶 ∈ Cat)
3 ciclcl 16673 . . . 4 ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐𝐶)𝐵) → 𝐴 ∈ (Base‘𝐶))
42, 3sylan 571 . . 3 ((𝜑𝐴( ≃𝑐𝐶)𝐵) → 𝐴 ∈ (Base‘𝐶))
5 cicrcl 16674 . . . 4 ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐𝐶)𝐵) → 𝐵 ∈ (Base‘𝐶))
62, 5sylan 571 . . 3 ((𝜑𝐴( ≃𝑐𝐶)𝐵) → 𝐵 ∈ (Base‘𝐶))
72adantr 468 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
8 cicsym 16675 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐𝐶)𝐵) → 𝐵( ≃𝑐𝐶)𝐴)
97, 8sylan 571 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐𝐶)𝐵) → 𝐵( ≃𝑐𝐶)𝐴)
10 eqid 2817 . . . . . . . . . . . 12 (Iso‘𝐶) = (Iso‘𝐶)
11 eqid 2817 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
12 simprr 780 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
13 simprl 778 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
1410, 11, 7, 12, 13cic 16670 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵( ≃𝑐𝐶)𝐴 ↔ ∃𝑘 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)))
15 initoeu1.a . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ (InitO‘𝐶))
16 eqid 2817 . . . . . . . . . . . . . . . . . 18 (Hom ‘𝐶) = (Hom ‘𝐶)
1711, 16, 2isinitoi 16864 . . . . . . . . . . . . . . . . 17 ((𝜑𝐴 ∈ (InitO‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)))
1815, 17mpdan 670 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)))
19 oveq2 6889 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑏 → (𝐴(Hom ‘𝐶)𝑎) = (𝐴(Hom ‘𝐶)𝑏))
2019eleq2d 2882 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑏 → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) ↔ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
2120eubidv 2647 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 𝑏 → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) ↔ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
2221rspcva 3511 . . . . . . . . . . . . . . . . . . . 20 ((𝑏 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏))
23 nfv 2005 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)
24 nfv 2005 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)
25 eleq1w 2879 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ∈ (𝐴(Hom ‘𝐶)𝑏)))
2623, 24, 25cbveu 2679 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ∃! ∈ (𝐴(Hom ‘𝐶)𝑏))
27 euex 2667 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃! ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃ ∈ (𝐴(Hom ‘𝐶)𝑏))
282adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
29 simpr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐵 ∈ (Base‘𝐶))
3029ad2antrl 710 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
31 simpl 470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐴 ∈ (Base‘𝐶))
3231ad2antrl 710 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
3311, 16, 10, 28, 30, 32isohom 16647 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → (𝐵(Iso‘𝐶)𝐴) ⊆ (𝐵(Hom ‘𝐶)𝐴))
3433sselda 3809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → 𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴))
35 eqid 2817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (comp‘𝐶) = (comp‘𝐶)
3628ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝐶 ∈ Cat)
3730ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝐵 ∈ (Base‘𝐶))
3832ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝐴 ∈ (Base‘𝐶))
39 simprr 780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝑏 ∈ (Base‘𝐶))
4039ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝑏 ∈ (Base‘𝐶))
41 simprl 778 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴))
42 simprr 780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) → ∈ (𝐴(Hom ‘𝐶)𝑏))
4311, 16, 35, 36, 37, 38, 40, 41, 42catcocl 16557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) → ((⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏))
44 simpl 470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝜑)
4544ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝜑)
4645adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ ((⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → 𝜑)
47 df-3an 1102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)) ↔ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)))
4847biimpri 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)))
4948ad4antlr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ ((⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)))
50 simpr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴))
5150ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ ((⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴))
5242adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ ((⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → ∈ (𝐴(Hom ‘𝐶)𝑏))
53 simpr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ ((⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → ((⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏))
542, 15, 11, 16, 10, 35initoeu2lem2 16876 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏) ∧ ((⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏))) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
5546, 49, 51, 52, 53, 54syl113anc 1494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ ((⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
5643, 55mpdan 670 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏))) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
5756ex 399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → ((𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ∈ (𝐴(Hom ‘𝐶)𝑏)) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
5834, 57mpand 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → ( ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
5958ex 399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → ( ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))
6059com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → ( ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))
6160ex 399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → ( ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
6261com15 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → ( ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
6362expd 402 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑏 ∈ (Base‘𝐶) → ( ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
6463com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ( ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
6564com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
6665exlimiv 2021 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃ ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
6727, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃! ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
6826, 67sylbi 208 . . . . . . . . . . . . . . . . . . . . . . 23 (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
6968pm2.43i 52 . . . . . . . . . . . . . . . . . . . . . 22 (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
7069com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 ∈ (Base‘𝐶) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
7170adantr 468 . . . . . . . . . . . . . . . . . . . 20 ((𝑏 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
7222, 71mpd 15 . . . . . . . . . . . . . . . . . . 19 ((𝑏 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))
7372ex 399 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
7473com15 101 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
7574adantld 480 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
7618, 75mpd 15 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))
7776imp 395 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
7877com12 32 . . . . . . . . . . . . 13 (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
7978exlimiv 2021 . . . . . . . . . . . 12 (∃𝑘 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
8079com12 32 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (∃𝑘 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
8114, 80sylbid 231 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵( ≃𝑐𝐶)𝐴 → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
8281adantr 468 . . . . . . . . 9 (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐𝐶)𝐵) → (𝐵( ≃𝑐𝐶)𝐴 → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
839, 82mpd 15 . . . . . . . 8 (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐𝐶)𝐵) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
8483an32s 634 . . . . . . 7 (((𝜑𝐴( ≃𝑐𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
8584imp 395 . . . . . 6 ((((𝜑𝐴( ≃𝑐𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑏 ∈ (Base‘𝐶)) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))
8685ralrimiva 3165 . . . . 5 (((𝜑𝐴( ≃𝑐𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))
872ad2antrr 708 . . . . . 6 (((𝜑𝐴( ≃𝑐𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
88 simprr 780 . . . . . 6 (((𝜑𝐴( ≃𝑐𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
8911, 16, 87, 88isinito 16861 . . . . 5 (((𝜑𝐴( ≃𝑐𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
9086, 89mpbird 248 . . . 4 (((𝜑𝐴( ≃𝑐𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (InitO‘𝐶))
9190ex 399 . . 3 ((𝜑𝐴( ≃𝑐𝐶)𝐵) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐵 ∈ (InitO‘𝐶)))
924, 6, 91mp2and 682 . 2 ((𝜑𝐴( ≃𝑐𝐶)𝐵) → 𝐵 ∈ (InitO‘𝐶))
931, 92mpdan 670 1 (𝜑𝐵 ∈ (InitO‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2157  ∃!weu 2641  wral 3107  cop 4387   class class class wbr 4855  cfv 6108  (class class class)co 6881  Basecbs 16075  Hom chom 16171  compcco 16172  Catccat 16536  Isociso 16617  𝑐 ccic 16666  InitOcinito 16849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5230  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-riota 6842  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-1st 7405  df-2nd 7406  df-supp 7537  df-cat 16540  df-cid 16541  df-sect 16618  df-inv 16619  df-iso 16620  df-cic 16667  df-inito 16852
This theorem is referenced by: (None)
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