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Theorem f1od 6067
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 6066 . 2  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
65simpld 112 1  |-  ( ph  ->  F : A -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    |-> cmpt 4061   `'ccnv 4621   -1-1-onto->wf1o 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218
This theorem is referenced by:  cnvf1o  6219  ixpsnf1o  6729  en2d  6761  mptfzshft  11421  fsumrev  11422  fprodrev  11598
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