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Theorem f1ocnvd 6537
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 )
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
f1ocnvd  |-  ( 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)    W( x, y)    X( x, y)

Proof of Theorem f1ocnvd
StepHypRef Expression
1 f1od.2 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  W )
21ralrimiva 2809 . . . 4  |-  ( ph  ->  A. x  e.  A  C  e.  W )
3 f1od.1 . . . . 5  |-  F  =  ( x  e.  A  |->  C )
43fnmpt 5714 . . . 4  |-  ( A. x  e.  A  C  e.  W  ->  F  Fn  A )
52, 4syl 17 . . 3  |-  ( ph  ->  F  Fn  A )
6 f1od.3 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  X )
76ralrimiva 2809 . . . . 5  |-  ( ph  ->  A. y  e.  B  D  e.  X )
8 eqid 2471 . . . . . 6  |-  ( y  e.  B  |->  D )  =  ( y  e.  B  |->  D )
98fnmpt 5714 . . . . 5  |-  ( A. y  e.  B  D  e.  X  ->  ( y  e.  B  |->  D )  Fn  B )
107, 9syl 17 . . . 4  |-  ( ph  ->  ( y  e.  B  |->  D )  Fn  B
)
11 f1od.4 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
1211opabbidv 4459 . . . . . 6  |-  ( ph  ->  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( y  e.  B  /\  x  =  D ) } )
13 df-mpt 4456 . . . . . . . . 9  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
143, 13eqtri 2493 . . . . . . . 8  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
1514cnveqi 5014 . . . . . . 7  |-  `' F  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
16 cnvopab 5243 . . . . . . 7  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }
1715, 16eqtri 2493 . . . . . 6  |-  `' F  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }
18 df-mpt 4456 . . . . . 6  |-  ( y  e.  B  |->  D )  =  { <. y ,  x >.  |  (
y  e.  B  /\  x  =  D ) }
1912, 17, 183eqtr4g 2530 . . . . 5  |-  ( ph  ->  `' F  =  (
y  e.  B  |->  D ) )
2019fneq1d 5676 . . . 4  |-  ( ph  ->  ( `' F  Fn  B 
<->  ( y  e.  B  |->  D )  Fn  B
) )
2110, 20mpbird 240 . . 3  |-  ( ph  ->  `' F  Fn  B
)
22 dff1o4 5836 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
235, 21, 22sylanbrc 677 . 2  |-  ( ph  ->  F : A -1-1-onto-> B )
2423, 19jca 541 1  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   {copab 4453    |-> cmpt 4454   `'ccnv 4838    Fn wfn 5584   -1-1-onto->wf1o 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596
This theorem is referenced by:  f1od  6538  f1ocnv2d  6539  pw2f1ocnv  35963
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