MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcnvs1 Structured version   Visualization version   Unicode version

Theorem funcnvs1 13066
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 5634 . 2  |-  Fun  `' { <. 0 ,  (  _I  `  A )
>. }
2 df-s1 12714 . . . 4  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
32cnveqi 5014 . . 3  |-  `' <" A ">  =  `' { <. 0 ,  (  _I  `  A )
>. }
43funeqi 5609 . 2  |-  ( Fun  `' <" A "> 
<->  Fun  `' { <. 0 ,  (  _I  `  A ) >. } )
51, 4mpbir 214 1  |-  Fun  `' <" A ">
Colors of variables: wff setvar class
Syntax hints:   {csn 3959   <.cop 3965    _I cid 4749   `'ccnv 4838   Fun wfun 5583   ` cfv 5589   0cc0 9557   <"cs1 12706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-fun 5591  df-s1 12714
This theorem is referenced by:  uhgr1wlkspthlem1  39945  1trld  40030
  Copyright terms: Public domain W3C validator