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Theorem f1ocnvd 6048
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 2543 . . . 4  |-  ( ph  ->  A. x  e.  A  C  e.  W )
3 f1od.1 . . . . 5  |-  F  =  ( x  e.  A  |->  C )
43fnmpt 5322 . . . 4  |-  ( A. x  e.  A  C  e.  W  ->  F  Fn  A )
52, 4syl 14 . . 3  |-  ( ph  ->  F  Fn  A )
6 f1od.3 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  X )
76ralrimiva 2543 . . . . 5  |-  ( ph  ->  A. y  e.  B  D  e.  X )
8 eqid 2170 . . . . . 6  |-  ( y  e.  B  |->  D )  =  ( y  e.  B  |->  D )
98fnmpt 5322 . . . . 5  |-  ( A. y  e.  B  D  e.  X  ->  ( y  e.  B  |->  D )  Fn  B )
107, 9syl 14 . . . 4  |-  ( ph  ->  ( y  e.  B  |->  D )  Fn  B
)
11 f1od.4 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
1211opabbidv 4053 . . . . . 6  |-  ( ph  ->  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( y  e.  B  /\  x  =  D ) } )
13 df-mpt 4050 . . . . . . . . 9  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
143, 13eqtri 2191 . . . . . . . 8  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
1514cnveqi 4784 . . . . . . 7  |-  `' F  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
16 cnvopab 5010 . . . . . . 7  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }
1715, 16eqtri 2191 . . . . . 6  |-  `' F  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }
18 df-mpt 4050 . . . . . 6  |-  ( y  e.  B  |->  D )  =  { <. y ,  x >.  |  (
y  e.  B  /\  x  =  D ) }
1912, 17, 183eqtr4g 2228 . . . . 5  |-  ( ph  ->  `' F  =  (
y  e.  B  |->  D ) )
2019fneq1d 5286 . . . 4  |-  ( ph  ->  ( `' F  Fn  B 
<->  ( y  e.  B  |->  D )  Fn  B
) )
2110, 20mpbird 166 . . 3  |-  ( ph  ->  `' F  Fn  B
)
22 dff1o4 5448 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
235, 21, 22sylanbrc 415 . 2  |-  ( ph  ->  F : A -1-1-onto-> B )
2423, 19jca 304 1  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   A.wral 2448   {copab 4047    |-> cmpt 4048   `'ccnv 4608    Fn wfn 5191   -1-1-onto->wf1o 5195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203
This theorem is referenced by:  f1od  6049  f1ocnv2d  6050
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