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Theorem funcnvs1 13877
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 6097 . 2 Fun {⟨0, ( I ‘𝐴)⟩}
2 df-s1 13508 . . . 4 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
32cnveqi 5452 . . 3 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
43funeqi 6070 . 2 (Fun ⟨“𝐴”⟩ ↔ Fun {⟨0, ( I ‘𝐴)⟩})
51, 4mpbir 221 1 Fun ⟨“𝐴”⟩
Colors of variables: wff setvar class
Syntax hints:  {csn 4321  cop 4327   I cid 5173  ccnv 5265  Fun wfun 6043  cfv 6049  0cc0 10148  ⟨“cs1 13500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-fun 6051  df-s1 13508
This theorem is referenced by:  uhgrwkspthlem1  26880  1trld  27315
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