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Theorem f1ocnvfv 5810
Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)
Assertion
Ref Expression
f1ocnvfv  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )

Proof of Theorem f1ocnvfv
StepHypRef Expression
1 fveq2 5541 . . 3  |-  ( D  =  ( F `  C )  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
21eqcoms 2299 . 2  |-  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
3 f1ocnvfv1 5808 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( `' F `  ( F `  C ) )  =  C )
43eqeq2d 2307 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( `' F `  D )  =  ( `' F `  ( F `
 C ) )  <-> 
( `' F `  D )  =  C ) )
52, 4syl5ib 210 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271
This theorem is referenced by:  f1ocnvfvb  5811  f1oiso2  5865  curry1  6226  curry2  6229  infxpenc2lem1  7662  axcclem  8099  uzrdgfni  11037  uzrdgsuci  11039  fzennn  11046  axdc4uzlem  11060  seqf1olem1  11101  seqf1olem2  11102  hashginv  11357  sadaddlem  12673  xpsaddlem  13493  xpsvsca  13497  xpsle  13499  catcisolem  13954  ghmf1o  14728  lmhmf1o  15819  symgtgp  17800  xpsdsval  17961  cnvbraval  22706  derangenlem  23717  subfacp1lem4  23729  subfacp1lem5  23730  cvmliftlem9  23839  trinv  25498  ltrinvlem  25509  rngoisocnv  26715  cdleme51finvfvN  31366  ltrniotacnvval  31393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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