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Theorem funcnvs1 14921
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 6609 . 2 Fun {⟨0, ( I ‘𝐴)⟩}
2 df-s1 14604 . . . 4 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
32cnveqi 5881 . . 3 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
43funeqi 6580 . 2 (Fun ⟨“𝐴”⟩ ↔ Fun {⟨0, ( I ‘𝐴)⟩})
51, 4mpbir 230 1 Fun ⟨“𝐴”⟩
Colors of variables: wff setvar class
Syntax hints:  {csn 4633  cop 4639   I cid 5579  ccnv 5681  Fun wfun 6548  cfv 6554  0cc0 11158  ⟨“cs1 14603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-fun 6556  df-s1 14604
This theorem is referenced by:  uhgrwkspthlem1  29690  1trld  30075
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