MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  initoeu1w Structured version   Visualization version   GIF version

Theorem initoeu1w 16863
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 16862 . . 3 (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5 euex 2631 . . 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 2760 . . 3 (Iso‘𝐶) = (Iso‘𝐶)
8 eqid 2760 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 initoo 16858 . . . 4 (𝐶 ∈ Cat → (𝐴 ∈ (InitO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
101, 2, 9sylc 65 . . 3 (𝜑𝐴 ∈ (Base‘𝐶))
11 initoo 16858 . . . 4 (𝐶 ∈ Cat → (𝐵 ∈ (InitO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
121, 3, 11sylc 65 . . 3 (𝜑𝐵 ∈ (Base‘𝐶))
137, 8, 1, 10, 12cic 16660 . 2 (𝜑 → (𝐴( ≃𝑐𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
146, 13mpbird 247 1 (𝜑𝐴( ≃𝑐𝐶)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1853  wcel 2139  ∃!weu 2607   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  Catccat 16526  Isociso 16607  𝑐 ccic 16656  InitOcinito 16839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-supp 7464  df-cat 16530  df-cid 16531  df-sect 16608  df-inv 16609  df-iso 16610  df-cic 16657  df-inito 16842
This theorem is referenced by:  nzerooringczr  42582
  Copyright terms: Public domain W3C validator