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Theorem f1ocnvfv 6159
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 5848 . . 3  |-  ( D  =  ( F `  C )  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
21eqcoms 2466 . 2  |-  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
3 f1ocnvfv1 6157 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( `' F `  ( F `  C ) )  =  C )
43eqeq2d 2468 . 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 367    = wceq 1398    e. wcel 1823   `'ccnv 4987   -1-1-onto->wf1o 5569   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578
This theorem is referenced by:  f1ocnvfvb  6160  f1oiso2  6223  curry1  6865  curry2  6868  mapfienlem2  7857  infxpenc2lem1  8387  axcclem  8828  uzrdgfni  12051  uzrdgsuci  12053  fzennn  12060  axdc4uzlem  12074  seqf1olem1  12128  seqf1olem2  12129  hashginv  12391  sadaddlem  14200  xpsaddlem  15064  xpsvsca  15068  xpsle  15070  catcisolem  15584  mhmf1o  16175  ghmf1o  16495  lmhmf1o  17887  symgtgp  20766  xpsdsval  21050  cnvbraval  27227  derangenlem  28879  subfacp1lem4  28891  subfacp1lem5  28892  cvmliftlem9  29002  rngoisocnv  30624  mgmhmf1o  32847  cdleme51finvfvN  36678  ltrniotacnvval  36705
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