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

Theorem initoeu1w 18000
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 (πœ‘ β†’ 𝐢 ∈ Cat)
initoeu1.a (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))
initoeu1.b (πœ‘ β†’ 𝐡 ∈ (InitOβ€˜πΆ))
Assertion
Ref Expression
initoeu1w (πœ‘ β†’ 𝐴( ≃𝑐 β€˜πΆ)𝐡)

Proof of Theorem initoeu1w
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 initoeu1.c . . . 4 (πœ‘ β†’ 𝐢 ∈ Cat)
2 initoeu1.a . . . 4 (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))
3 initoeu1.b . . . 4 (πœ‘ β†’ 𝐡 ∈ (InitOβ€˜πΆ))
41, 2, 3initoeu1 17999 . . 3 (πœ‘ β†’ βˆƒ!𝑓 𝑓 ∈ (𝐴(Isoβ€˜πΆ)𝐡))
5 euex 2565 . . 3 (βˆƒ!𝑓 𝑓 ∈ (𝐴(Isoβ€˜πΆ)𝐡) β†’ βˆƒπ‘“ 𝑓 ∈ (𝐴(Isoβ€˜πΆ)𝐡))
64, 5syl 17 . 2 (πœ‘ β†’ βˆƒπ‘“ 𝑓 ∈ (𝐴(Isoβ€˜πΆ)𝐡))
7 eqid 2725 . . 3 (Isoβ€˜πΆ) = (Isoβ€˜πΆ)
8 eqid 2725 . . 3 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
9 initoo 17995 . . . 4 (𝐢 ∈ Cat β†’ (𝐴 ∈ (InitOβ€˜πΆ) β†’ 𝐴 ∈ (Baseβ€˜πΆ)))
101, 2, 9sylc 65 . . 3 (πœ‘ β†’ 𝐴 ∈ (Baseβ€˜πΆ))
11 initoo 17995 . . . 4 (𝐢 ∈ Cat β†’ (𝐡 ∈ (InitOβ€˜πΆ) β†’ 𝐡 ∈ (Baseβ€˜πΆ)))
121, 3, 11sylc 65 . . 3 (πœ‘ β†’ 𝐡 ∈ (Baseβ€˜πΆ))
137, 8, 1, 10, 12cic 17781 . 2 (πœ‘ β†’ (𝐴( ≃𝑐 β€˜πΆ)𝐡 ↔ βˆƒπ‘“ 𝑓 ∈ (𝐴(Isoβ€˜πΆ)𝐡)))
146, 13mpbird 256 1 (πœ‘ β†’ 𝐴( ≃𝑐 β€˜πΆ)𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4  βˆƒwex 1773   ∈ wcel 2098  βˆƒ!weu 2556   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179  Catccat 17643  Isociso 17728   ≃𝑐 ccic 17777  InitOcinito 17969
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 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
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 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-supp 8164  df-cat 17647  df-cid 17648  df-sect 17729  df-inv 17730  df-iso 17731  df-cic 17778  df-inito 17972
This theorem is referenced by:  nzerooringczr  21410
  Copyright terms: Public domain W3C validator