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Theorem f1ocnvfvb 5743
Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
Assertion
Ref Expression
f1ocnvfvb  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  <-> 
( `' F `  D )  =  C ) )

Proof of Theorem f1ocnvfvb
StepHypRef Expression
1 f1ocnvfv 5742 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
213adant3 1006 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  ->  ( `' F `  D )  =  C ) )
3 fveq2 5481 . . . . 5  |-  ( C  =  ( `' F `  D )  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
43eqcoms 2167 . . . 4  |-  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  ( F `  ( `' F `  D ) ) )
5 f1ocnvfv2 5741 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( F `  ( `' F `  D ) )  =  D )
65eqeq2d 2176 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( ( F `  C )  =  ( F `  ( `' F `  D ) )  <->  ( F `  C )  =  D ) )
74, 6syl5ib 153 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  D  e.  B )  ->  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  D ) )
873adant2 1005 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( `' F `  D )  =  C  ->  ( F `  C )  =  D ) )
92, 8impbid 128 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( ( F `  C )  =  D  <-> 
( `' F `  D )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 967    = wceq 1342    e. wcel 2135   `'ccnv 4598   -1-1-onto->wf1o 5182   ` cfv 5183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-sbc 2948  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-id 4266  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191
This theorem is referenced by:  f1ofveu  5825  f1ocnvfv3  5826  seq3f1olemstep  10427  1arith2  12287  ennnfonelem1  12303  txhmeo  12886
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