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Theorem f1ocnvd 6285
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 2781 . . . 4  |-  ( ph  ->  A. x  e.  A  C  e.  W )
3 f1od.1 . . . . 5  |-  F  =  ( x  e.  A  |->  C )
43fnmpt 5563 . . . 4  |-  ( A. x  e.  A  C  e.  W  ->  F  Fn  A )
52, 4syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
6 f1od.3 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  X )
76ralrimiva 2781 . . . . 5  |-  ( ph  ->  A. y  e.  B  D  e.  X )
8 eqid 2435 . . . . . 6  |-  ( y  e.  B  |->  D )  =  ( y  e.  B  |->  D )
98fnmpt 5563 . . . . 5  |-  ( A. y  e.  B  D  e.  X  ->  ( y  e.  B  |->  D )  Fn  B )
107, 9syl 16 . . . 4  |-  ( ph  ->  ( y  e.  B  |->  D )  Fn  B
)
11 f1od.4 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
1211opabbidv 4263 . . . . . 6  |-  ( ph  ->  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( y  e.  B  /\  x  =  D ) } )
13 df-mpt 4260 . . . . . . . . 9  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
143, 13eqtri 2455 . . . . . . . 8  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
1514cnveqi 5039 . . . . . . 7  |-  `' F  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
16 cnvopab 5266 . . . . . . 7  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }
1715, 16eqtri 2455 . . . . . 6  |-  `' F  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }
18 df-mpt 4260 . . . . . 6  |-  ( y  e.  B  |->  D )  =  { <. y ,  x >.  |  (
y  e.  B  /\  x  =  D ) }
1912, 17, 183eqtr4g 2492 . . . . 5  |-  ( ph  ->  `' F  =  (
y  e.  B  |->  D ) )
2019fneq1d 5528 . . . 4  |-  ( ph  ->  ( `' F  Fn  B 
<->  ( y  e.  B  |->  D )  Fn  B
) )
2110, 20mpbird 224 . . 3  |-  ( ph  ->  `' F  Fn  B
)
22 dff1o4 5674 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
235, 21, 22sylanbrc 646 . 2  |-  ( ph  ->  F : A -1-1-onto-> B )
2423, 19jca 519 1  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {copab 4257    e. cmpt 4258   `'ccnv 4869    Fn wfn 5441   -1-1-onto->wf1o 5445
This theorem is referenced by:  f1od  6286  f1ocnv2d  6287  pw2f1ocnv  27099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453
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