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Theorem f1ocnvfv 5982
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 5688 . . 3  |-  ( D  =  ( F `  C )  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
21eqcoms 2444 . 2  |-  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
3 f1ocnvfv1 5980 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( `' F `  ( F `  C ) )  =  C )
43eqeq2d 2452 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( `' F `  D )  =  ( `' F `  ( F `
 C ) )  <-> 
( `' F `  D )  =  C ) )
52, 4syl5ib 219 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   `'ccnv 4835   -1-1-onto->wf1o 5414   ` cfv 5415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423
This theorem is referenced by:  f1ocnvfvb  5983  f1oiso2  6040  curry1  6663  curry2  6666  mapfienlem2  7651  infxpenc2lem1  8181  axcclem  8622  uzrdgfni  11777  uzrdgsuci  11779  fzennn  11786  axdc4uzlem  11800  seqf1olem1  11841  seqf1olem2  11842  hashginv  12103  sadaddlem  13658  xpsaddlem  14509  xpsvsca  14513  xpsle  14515  catcisolem  14970  ghmf1o  15769  lmhmf1o  17105  symgtgp  19631  xpsdsval  19915  cnvbraval  25449  derangenlem  26989  subfacp1lem4  27001  subfacp1lem5  27002  cvmliftlem9  27112  rngoisocnv  28712  cdleme51finvfvN  33921  ltrniotacnvval  33948
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