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Theorem f1od 6083
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 6082 . 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:  cnvf1o  6233  ixpsnf1o  6872  en2d  6913  pw2f1o  6983  seqf1olem1  11101
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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