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Theorem f1ocnv2d 6510
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 2550 . . . . . 6  |-  ( C  e.  B  ->  (
y  =  C  -> 
y  e.  B ) )
52, 4syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  C  -> 
y  e.  B ) )
65impr 619 . . . 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 485 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  y  =  C )  ->  x  =  D )
98exp42 611 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( y  =  C  ->  x  =  D ) ) ) )
109com34 83 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  ( y  =  C  ->  ( y  e.  B  ->  x  =  D ) ) ) )
1110imp32 433 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  ->  x  =  D ) )
126, 11jcai 536 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  /\  x  =  D
) )
13 eleq1a 2550 . . . . . 6  |-  ( D  e.  A  ->  (
x  =  D  ->  x  e.  A )
)
143, 13syl 16 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
x  =  D  ->  x  e.  A )
)
1514impr 619 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  ->  x  e.  A )
167biimpa 484 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  x  =  D )  ->  y  =  C )
1716exp42 611 . . . . . . 7  |-  ( ph  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( x  =  D  ->  y  =  C ) ) ) )
1817com23 78 . . . . . 6  |-  ( ph  ->  ( y  e.  B  ->  ( x  e.  A  ->  ( x  =  D  ->  y  =  C ) ) ) )
1918com34 83 . . . . 5  |-  ( ph  ->  ( y  e.  B  ->  ( x  =  D  ->  ( x  e.  A  ->  y  =  C ) ) ) )
2019imp32 433 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  -> 
( x  e.  A  ->  y  =  C ) )
2115, 20jcai 536 . . 3  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  -> 
( x  e.  A  /\  y  =  C
) )
2212, 21impbida 830 . 2  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
231, 2, 3, 22f1ocnvd 6508 1  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505   `'ccnv 4998   -1-1-onto->wf1o 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595
This theorem is referenced by:  f1o2d  6511  negiso  10519  iccf1o  11664  bitsf1ocnv  13953  grpinvcnv  15916  grplactcnv  15948  issrngd  17310  opncldf1  19379  txhmeo  20067  ptuncnv  20071  icopnfcnv  21205  iccpnfcnv  21207  xrge0iifcnv  27579
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