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Theorem funcnvs1 13507
 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 5850 . 2 Fun {⟨0, ( I ‘𝐴)⟩}
2 df-s1 13157 . . . 4 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
32cnveqi 5219 . . 3 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
43funeqi 5824 . 2 (Fun ⟨“𝐴”⟩ ↔ Fun {⟨0, ( I ‘𝐴)⟩})
51, 4mpbir 220 1 Fun ⟨“𝐴”⟩
 Colors of variables: wff setvar class Syntax hints:  {csn 4125  ⟨cop 4131   I cid 4948  ◡ccnv 5037  Fun wfun 5798  ‘cfv 5804  0cc0 9815  ⟨“cs1 13149 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806  df-s1 13157 This theorem is referenced by:  uhgr1wlkspthlem1  40959  1trld  41309
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