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Theorem f1ocnv2d 6297
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 2507 . . . . . 6  |-  ( C  e.  B  ->  (
y  =  C  -> 
y  e.  B ) )
52, 4syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  C  -> 
y  e.  B ) )
65impr 604 . . . 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 473 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  y  =  C )  ->  x  =  D )
98exp42 596 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( y  =  C  ->  x  =  D ) ) ) )
109com34 80 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  ( y  =  C  ->  ( y  e.  B  ->  x  =  D ) ) ) )
1110imp32 424 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  ->  x  =  D ) )
126, 11jcai 524 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  /\  x  =  D
) )
13 eleq1a 2507 . . . . . 6  |-  ( D  e.  A  ->  (
x  =  D  ->  x  e.  A )
)
143, 13syl 16 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
x  =  D  ->  x  e.  A )
)
1514impr 604 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  ->  x  e.  A )
167biimpa 472 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  x  =  D )  ->  y  =  C )
1716exp42 596 . . . . . . 7  |-  ( ph  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( x  =  D  ->  y  =  C ) ) ) )
1817com23 75 . . . . . 6  |-  ( ph  ->  ( y  e.  B  ->  ( x  e.  A  ->  ( x  =  D  ->  y  =  C ) ) ) )
1918com34 80 . . . . 5  |-  ( ph  ->  ( y  e.  B  ->  ( x  =  D  ->  ( x  e.  A  ->  y  =  C ) ) ) )
2019imp32 424 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  -> 
( x  e.  A  ->  y  =  C ) )
2115, 20jcai 524 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  -> 
( x  e.  A  /\  y  =  C
) )
2212, 21impbida 807 . 2  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
231, 2, 3, 22f1ocnvd 6295 1  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4268   `'ccnv 4879   -1-1-onto->wf1o 5455
This theorem is referenced by:  f1o2d  6298  negiso  9986  iccf1o  11041  bitsf1ocnv  12958  grpinvcnv  14861  grplactcnv  14889  issrngd  15951  opncldf1  17150  txhmeo  17837  ptuncnv  17841  icopnfcnv  18969  iccpnfcnv  18971  xrge0iifcnv  24321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463
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