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Theorem f1o2d 6085
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
Hypotheses
Ref Expression
f1od.1  |-  F  =  ( x  e.  A  |->  C )
f1o2d.2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
f1o2d.3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
f1o2d.4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
Assertion
Ref Expression
f1o2d  |-  ( ph  ->  F : A -1-1-onto-> B )
Distinct variable groups:    x, y, A    x, B, y    y, C    x, D    ph, x, y
Allowed substitution hints:    C( x)    D( y)    F( x, y)

Proof of Theorem f1o2d
StepHypRef Expression
1 f1od.1 . . 3  |-  F  =  ( x  e.  A  |->  C )
2 f1o2d.2 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
3 f1o2d.3 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
4 f1o2d.4 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
51, 2, 3, 4f1ocnv2d 6084 . 2  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
65simpld 445 1  |-  ( ph  ->  F : A -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   `'ccnv 4704   -1-1-onto->wf1o 5270
This theorem is referenced by:  f1opw2  6087  en3d  6914  f1opwfi  7175  mapfien  7415  fin23lem22  7969  incexclem  12311  grplmulf1o  14558  conjghm  14729  gapm  14776  psrbagconf1o  16136  hmeoimaf1o  17477  itg1mulc  19075  resinf1o  19914  eff1olem  19926  sqff1o  20436  dvdsflip  20438  dvdsppwf1o  20442  dvdsflf1o  20443  cbicp  25269  hashgcdlem  27619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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