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Theorem f1ocnvfv 6008
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 5720 . . 3  |-  ( D  =  ( F `  C )  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
21eqcoms 2438 . 2  |-  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
3 f1ocnvfv1 6006 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( `' F `  ( F `  C ) )  =  C )
43eqeq2d 2446 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( `' F `  D )  =  ( `' F `  ( F `
 C ) )  <-> 
( `' F `  D )  =  C ) )
52, 4syl5ib 211 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 359    = wceq 1652    e. wcel 1725   `'ccnv 4869   -1-1-onto->wf1o 5445   ` cfv 5446
This theorem is referenced by:  f1ocnvfvb  6009  f1oiso2  6064  curry1  6430  curry2  6433  infxpenc2lem1  7892  axcclem  8329  uzrdgfni  11290  uzrdgsuci  11292  fzennn  11299  axdc4uzlem  11313  seqf1olem1  11354  seqf1olem2  11355  hashginv  11614  sadaddlem  12970  xpsaddlem  13792  xpsvsca  13796  xpsle  13798  catcisolem  14253  ghmf1o  15027  lmhmf1o  16114  symgtgp  18123  xpsdsval  18403  cnvbraval  23605  derangenlem  24849  subfacp1lem4  24861  subfacp1lem5  24862  cvmliftlem9  24972  rngoisocnv  26588  cdleme51finvfvN  31289  ltrniotacnvval  31316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454
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