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Theorem f1o2d 5736
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 5735 . 2  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
65simpld 110 1  |-  ( ph  ->  F : A -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434    |-> cmpt 3847   `'ccnv 4370   -1-1-onto->wf1o 4931
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939
This theorem is referenced by:  f1opw2  5737  en3d  6316  fidifsnen  6405  dvdsflip  10396
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