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Theorem funcnvs1 14952
Description: The converse of a singleton word is a function. (Contributed by AV, 22-Jan-2021.)
Assertion
Ref Expression
funcnvs1 Fun ⟨“𝐴”⟩

Proof of Theorem funcnvs1
StepHypRef Expression
1 funcnvsn 6615 . 2 Fun {⟨0, ( I ‘𝐴)⟩}
2 df-s1 14635 . . . 4 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
32cnveqi 5884 . . 3 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
43funeqi 6586 . 2 (Fun ⟨“𝐴”⟩ ↔ Fun {⟨0, ( I ‘𝐴)⟩})
51, 4mpbir 231 1 Fun ⟨“𝐴”⟩
Colors of variables: wff setvar class
Syntax hints:  {csn 4625  cop 4631   I cid 5576  ccnv 5683  Fun wfun 6554  cfv 6560  0cc0 11156  ⟨“cs1 14634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-fun 6562  df-s1 14635
This theorem is referenced by:  uhgrwkspthlem1  29774  1trld  30162
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