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Theorem f1ocnvd 5730
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 2409 . . . 4  |-  ( ph  ->  A. x  e.  A  C  e.  W )
3 f1od.1 . . . . 5  |-  F  =  ( x  e.  A  |->  C )
43fnmpt 5053 . . . 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 2409 . . . . 5  |-  ( ph  ->  A. y  e.  B  D  e.  X )
8 eqid 2056 . . . . . 6  |-  ( y  e.  B  |->  D )  =  ( y  e.  B  |->  D )
98fnmpt 5053 . . . . 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 3851 . . . . . 6  |-  ( ph  ->  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( y  e.  B  /\  x  =  D ) } )
13 df-mpt 3848 . . . . . . . . 9  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
143, 13eqtri 2076 . . . . . . . 8  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
1514cnveqi 4538 . . . . . . 7  |-  `' F  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }
16 cnvopab 4754 . . . . . . 7  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }
1715, 16eqtri 2076 . . . . . 6  |-  `' F  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  C ) }
18 df-mpt 3848 . . . . . 6  |-  ( y  e.  B  |->  D )  =  { <. y ,  x >.  |  (
y  e.  B  /\  x  =  D ) }
1912, 17, 183eqtr4g 2113 . . . . 5  |-  ( ph  ->  `' F  =  (
y  e.  B  |->  D ) )
2019fneq1d 5017 . . . 4  |-  ( ph  ->  ( `' F  Fn  B 
<->  ( y  e.  B  |->  D )  Fn  B
) )
2110, 20mpbird 160 . . 3  |-  ( ph  ->  `' F  Fn  B
)
22 dff1o4 5162 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
235, 21, 22sylanbrc 402 . 2  |-  ( ph  ->  F : A -1-1-onto-> B )
2423, 19jca 294 1  |-  ( ph  ->  ( F : A -1-1-onto-> B  /\  `' F  =  (
y  e.  B  |->  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   A.wral 2323   {copab 3845    |-> cmpt 3846   `'ccnv 4372    Fn wfn 4925   -1-1-onto->wf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  f1od  5731  f1ocnv2d  5732
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