Theorem initoeu1w 16886

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 16885	. . 3 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5		euex 2667	. . 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 2817	. . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8		eqid 2817	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		initoo 16881	. . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (InitO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
10	1, 2, 9	sylc 65	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶))
11		initoo 16881	. . . 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 16683	. 2 ⊢ (𝜑 → (𝐴( ≃𝑐 ‘𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
14	6, 13	mpbird 248	1 ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)

Syntax hints:   → wi 4  ∃wex 1859   ∈ wcel 2157  ∃!weu 2641   class class class wbr 4855  ‘cfv 6111  (class class class)co 6884  Basecbs 16088  Catccat 16549  Isociso 16630   ≃_𝑐 ccic 16679  InitOcinito 16862

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 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189

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 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-1st 7408  df-2nd 7409  df-supp 7540  df-cat 16553  df-cid 16554  df-sect 16631  df-inv 16632  df-iso 16633  df-cic 16680  df-inito 16865

This theorem is referenced by:  nzerooringczr  42658