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Theorem relcnvfi 6648
Description: If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)
Assertion
Ref Expression
relcnvfi  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' A  e.  Fin )

Proof of Theorem relcnvfi
StepHypRef Expression
1 dfrel2 4881 . . . . 5  |-  ( Rel 
A  <->  `' `' A  =  A
)
21biimpi 118 . . . 4  |-  ( Rel 
A  ->  `' `' A  =  A )
32adantr 270 . . 3  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' `' A  =  A
)
4 simpr 108 . . 3  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  A  e.  Fin )
53, 4eqeltrd 2164 . 2  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' `' A  e.  Fin )
6 relcnv 4810 . . . 4  |-  Rel  `' A
7 cnvexg 4968 . . . 4  |-  ( A  e.  Fin  ->  `' A  e.  _V )
8 cnven 6523 . . . 4  |-  ( ( Rel  `' A  /\  `' A  e.  _V )  ->  `' A  ~~  `' `' A )
96, 7, 8sylancr 405 . . 3  |-  ( A  e.  Fin  ->  `' A  ~~  `' `' A
)
109adantl 271 . 2  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' A  ~~  `' `' A
)
11 enfii 6588 . 2  |-  ( ( `' `' A  e.  Fin  /\  `' A  ~~  `' `' A )  ->  `' A  e.  Fin )
125, 10, 11syl2anc 403 1  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' A  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619   class class class wbr 3845   `'ccnv 4437   Rel wrel 4443    ~~ cen 6453   Fincfn 6455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-er 6290  df-en 6456  df-fin 6458
This theorem is referenced by:  funrnfi  6649  fsumcnv  10827
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