ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1ocnvfv Unicode version

Theorem f1ocnvfv 5558
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 5305 . . 3  |-  ( D  =  ( F `  C )  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
21eqcoms 2091 . 2  |-  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  ( `' F `  ( F `  C
) ) )
3 f1ocnvfv1 5556 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( `' F `  ( F `  C ) )  =  C )
43eqeq2d 2099 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( `' F `  D )  =  ( `' F `  ( F `
 C ) )  <-> 
( `' F `  D )  =  C ) )
52, 4syl5ib 152 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 102    = wceq 1289    e. wcel 1438   `'ccnv 4437   -1-1-onto->wf1o 5014   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023
This theorem is referenced by:  f1ocnvfvb  5559  f1oiso2  5606  frecuzrdgtcl  9819  frecuzrdgsuc  9821  frecuzrdgfunlem  9826  frecfzennn  9833  0tonninf  9845  1tonninf  9846  sqpweven  11431  2sqpwodd  11432
  Copyright terms: Public domain W3C validator