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Theorem f1od 5976
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 )
f1od.2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  W )
f1od.3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  X )
f1od.4  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
Assertion
Ref Expression
f1od  |-  ( 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)    W( x, y)    X( x, y)

Proof of Theorem f1od
StepHypRef Expression
1 f1od.1 . . 3  |-  F  =  ( x  e.  A  |->  C )
2 f1od.2 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  W )
3 f1od.3 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  X )
4 f1od.4 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
51, 2, 3, 4f1ocnvd 5975 . 2  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
65simpld 111 1  |-  ( ph  ->  F : A -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480    |-> cmpt 3992   `'ccnv 4541   -1-1-onto->wf1o 5125
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133
This theorem is referenced by:  cnvf1o  6125  ixpsnf1o  6633  en2d  6665  mptfzshft  11235  fsumrev  11236
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