Theorem initoeu1w 18057

Description: Initial objects are essentially unique (weak form), i.e. if A and B are initial objects, then A and B are isomorphic. Proposition 7.3 (1) of [Adamek] p. 102. (Contributed by AV, 6-Apr-2020.)

Hypotheses

Ref	Expression
initoeu1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
initoeu1.a	⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
initoeu1.b	⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))

Assertion

Ref	Expression
initoeu1w	⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)

Proof of Theorem initoeu1w

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		initoeu1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		initoeu1.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
3		initoeu1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))
4	1, 2, 3	initoeu1 18056	. . 3 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5		euex 2577	. . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
7		eqid 2737	. . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8		eqid 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		initoo 18052	. . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (InitO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
10	1, 2, 9	sylc 65	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶))
11		initoo 18052	. . . 4 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (InitO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
12	1, 3, 11	sylc 65	. . 3 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶))
13	7, 8, 1, 10, 12	cic 17843	. 2 ⊢ (𝜑 → (𝐴( ≃𝑐 ‘𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
14	6, 13	mpbird 257	1 ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)

Syntax hints:   → wi 4  ∃wex 1779   ∈ wcel 2108  ∃!weu 2568   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Catccat 17707  Isociso 17790   ≃_𝑐 ccic 17839  InitOcinito 18026

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-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-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  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-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-1st 8014  df-2nd 8015  df-supp 8186  df-cat 17711  df-cid 17712  df-sect 17791  df-inv 17792  df-iso 17793  df-cic 17840  df-inito 18029

This theorem is referenced by:  nzerooringczr  21491