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Theorem relcnvfi 6916
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 5059 . . . . 5  |-  ( Rel 
A  <->  `' `' A  =  A
)
21biimpi 119 . . . 4  |-  ( Rel 
A  ->  `' `' A  =  A )
32adantr 274 . . 3  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' `' A  =  A
)
4 simpr 109 . . 3  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  A  e.  Fin )
53, 4eqeltrd 2247 . 2  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' `' A  e.  Fin )
6 relcnv 4987 . . . 4  |-  Rel  `' A
7 cnvexg 5146 . . . 4  |-  ( A  e.  Fin  ->  `' A  e.  _V )
8 cnven 6784 . . . 4  |-  ( ( Rel  `' A  /\  `' A  e.  _V )  ->  `' A  ~~  `' `' A )
96, 7, 8sylancr 412 . . 3  |-  ( A  e.  Fin  ->  `' A  ~~  `' `' A
)
109adantl 275 . 2  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' A  ~~  `' `' A
)
11 enfii 6850 . 2  |-  ( ( `' `' A  e.  Fin  /\  `' A  ~~  `' `' A )  ->  `' A  e.  Fin )
125, 10, 11syl2anc 409 1  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' A  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3987   `'ccnv 4608   Rel wrel 4614    ~~ cen 6714   Fincfn 6716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1st 6117  df-2nd 6118  df-er 6511  df-en 6717  df-fin 6719
This theorem is referenced by:  funrnfi  6917  fsumcnv  11393  fprodcnv  11581
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