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Theorem f1ocnv2d 6499
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 2537 . . . . . 6  |-  ( C  e.  B  ->  (
y  =  C  -> 
y  e.  B ) )
52, 4syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  C  -> 
y  e.  B ) )
65impr 617 . . . 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 483 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  y  =  C )  ->  x  =  D )
98exp42 609 . . . . . 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 431 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  ->  x  =  D ) )
126, 11jcai 534 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  =  C ) )  -> 
( y  e.  B  /\  x  =  D
) )
13 eleq1a 2537 . . . . . 6  |-  ( D  e.  A  ->  (
x  =  D  ->  x  e.  A )
)
143, 13syl 16 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  (
x  =  D  ->  x  e.  A )
)
1514impr 617 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  ->  x  e.  A )
167biimpa 482 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  B )
)  /\  x  =  D )  ->  y  =  C )
1716exp42 609 . . . . . . 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 431 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  x  =  D ) )  -> 
( x  e.  A  ->  y  =  C ) )
2115, 20jcai 534 . . 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 6497 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 367    = wceq 1398    e. wcel 1823    |-> cmpt 4497   `'ccnv 4987   -1-1-onto->wf1o 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577
This theorem is referenced by:  f1o2d  6500  negiso  10514  iccf1o  11667  bitsf1ocnv  14178  grpinvcnv  16305  grplactcnv  16337  issrngd  17705  opncldf1  19752  txhmeo  20470  ptuncnv  20474  icopnfcnv  21608  iccpnfcnv  21610  xrge0iifcnv  28150
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