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Theorem initoeu1w 16431
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 (𝜑𝐶 ∈ Cat)
initoeu1.a (𝜑𝐴 ∈ (InitO‘𝐶))
initoeu1.b (𝜑𝐵 ∈ (InitO‘𝐶))
Assertion
Ref Expression
initoeu1w (𝜑𝐴( ≃𝑐𝐶)𝐵)

Proof of Theorem initoeu1w
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 initoeu1.c . . . 4 (𝜑𝐶 ∈ Cat)
2 initoeu1.a . . . 4 (𝜑𝐴 ∈ (InitO‘𝐶))
3 initoeu1.b . . . 4 (𝜑𝐵 ∈ (InitO‘𝐶))
41, 2, 3initoeu1 16430 . . 3 (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5 euex 2481 . . 3 (∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
64, 5syl 17 . 2 (𝜑 → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
7 eqid 2609 . . 3 (Iso‘𝐶) = (Iso‘𝐶)
8 eqid 2609 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 initoo 16426 . . . 4 (𝐶 ∈ Cat → (𝐴 ∈ (InitO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
101, 2, 9sylc 62 . . 3 (𝜑𝐴 ∈ (Base‘𝐶))
11 initoo 16426 . . . 4 (𝐶 ∈ Cat → (𝐵 ∈ (InitO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
121, 3, 11sylc 62 . . 3 (𝜑𝐵 ∈ (Base‘𝐶))
137, 8, 1, 10, 12cic 16228 . 2 (𝜑 → (𝐴( ≃𝑐𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
146, 13mpbird 245 1 (𝜑𝐴( ≃𝑐𝐶)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1694  wcel 1976  ∃!weu 2457   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  Catccat 16094  Isociso 16175  𝑐 ccic 16224  InitOcinito 16407
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 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  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-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-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  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-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-1st 7036  df-2nd 7037  df-supp 7160  df-cat 16098  df-cid 16099  df-sect 16176  df-inv 16177  df-iso 16178  df-cic 16225  df-inito 16410
This theorem is referenced by:  nzerooringczr  41859
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