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Theorem f1ocnv2d 6084
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
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
f1ocnv2d  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
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 f1ocnv2d
StepHypRef Expression
1 f1od.1 . 2  |-  F  =  ( x  e.  A  |->  C )
2 f1o2d.2 . 2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
3 f1o2d.3 . 2  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
4 eleq1a 2365 . . . . . 6  |-  ( C  e.  B  ->  (
y  =  C  -> 
y  e.  B ) )
52, 4syl 15 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  C  -> 
y  e.  B ) )
65impr 602 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
y  e.  B )
7 f1o2d.4 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
87biimpar 471 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  y  =  C )  ->  x  =  D )
98exp42 594 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( y  =  C  ->  x  =  D ) ) ) )
109com34 77 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  ( y  =  C  ->  ( y  e.  B  ->  x  =  D ) ) ) )
1110imp32 422 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  ->  x  =  D ) )
126, 11jcai 522 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  /\  x  =  D
) )
13 eleq1a 2365 . . . . . 6  |-  ( D  e.  A  ->  (
x  =  D  ->  x  e.  A )
)
143, 13syl 15 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
x  =  D  ->  x  e.  A )
)
1514impr 602 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  ->  x  e.  A )
167biimpa 470 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  x  =  D )  ->  y  =  C )
1716exp42 594 . . . . . . 7  |-  ( ph  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( x  =  D  ->  y  =  C ) ) ) )
1817com23 72 . . . . . 6  |-  ( ph  ->  ( y  e.  B  ->  ( x  e.  A  ->  ( x  =  D  ->  y  =  C ) ) ) )
1918com34 77 . . . . 5  |-  ( ph  ->  ( y  e.  B  ->  ( x  =  D  ->  ( x  e.  A  ->  y  =  C ) ) ) )
2019imp32 422 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  -> 
( x  e.  A  ->  y  =  C ) )
2115, 20jcai 522 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  -> 
( x  e.  A  /\  y  =  C
) )
2212, 21impbida 805 . 2  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
231, 2, 3, 22f1ocnvd 6082 1  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
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:  f1o2d  6085  negiso  9746  iccf1o  10794  bitsf1ocnv  12651  grpinvcnv  14552  grplactcnv  14580  issrngd  15642  opncldf1  16837  txhmeo  17510  ptuncnv  17514  icopnfcnv  18456  iccpnfcnv  18458  xrge0iifcnv  23330  cnvtr  25719
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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