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Theorem initoeu1w 18004
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 18003 . . 3 (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
5 euex 2565 . . 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 2725 . . 3 (Iso‘𝐶) = (Iso‘𝐶)
8 eqid 2725 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 initoo 17999 . . . 4 (𝐶 ∈ Cat → (𝐴 ∈ (InitO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
101, 2, 9sylc 65 . . 3 (𝜑𝐴 ∈ (Base‘𝐶))
11 initoo 17999 . . . 4 (𝐶 ∈ Cat → (𝐵 ∈ (InitO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
121, 3, 11sylc 65 . . 3 (𝜑𝐵 ∈ (Base‘𝐶))
137, 8, 1, 10, 12cic 17785 . 2 (𝜑 → (𝐴( ≃𝑐𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
146, 13mpbird 256 1 (𝜑𝐴( ≃𝑐𝐶)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1773  wcel 2098  ∃!weu 2556   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  Catccat 17647  Isociso 17732  𝑐 ccic 17781  InitOcinito 17973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-supp 8166  df-cat 17651  df-cid 17652  df-sect 17733  df-inv 17734  df-iso 17735  df-cic 17782  df-inito 17976
This theorem is referenced by:  nzerooringczr  21423
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