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Theorem pmvalg 6519
Description: The value of the partial mapping operation.  ( A  ^pm  B ) is the set of all partial functions that map from  B to  A. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
pmvalg  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  ^pm  B
)  =  { f  e.  ~P ( B  X.  A )  |  Fun  f } )
Distinct variable groups:    A, f    B, f
Allowed substitution hints:    C( f)    D( f)

Proof of Theorem pmvalg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3150 . . 3  |-  { f  e.  ~P ( B  X.  A )  |  Fun  f }  C_  ~P ( B  X.  A
)
2 xpexg 4621 . . . . 5  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( B  X.  A
)  e.  _V )
32ancoms 266 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( B  X.  A
)  e.  _V )
4 pwexg 4072 . . . 4  |-  ( ( B  X.  A )  e.  _V  ->  ~P ( B  X.  A
)  e.  _V )
53, 4syl 14 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ~P ( B  X.  A )  e.  _V )
6 ssexg 4035 . . 3  |-  ( ( { f  e.  ~P ( B  X.  A
)  |  Fun  f }  C_  ~P ( B  X.  A )  /\  ~P ( B  X.  A
)  e.  _V )  ->  { f  e.  ~P ( B  X.  A
)  |  Fun  f }  e.  _V )
71, 5, 6sylancr 408 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  e.  ~P ( B  X.  A
)  |  Fun  f }  e.  _V )
8 elex 2669 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
9 elex 2669 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
10 xpeq2 4522 . . . . . . 7  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
1110pweqd 3483 . . . . . 6  |-  ( x  =  A  ->  ~P ( y  X.  x
)  =  ~P (
y  X.  A ) )
12 rabeq 2650 . . . . . 6  |-  ( ~P ( y  X.  x
)  =  ~P (
y  X.  A )  ->  { f  e. 
~P ( y  X.  x )  |  Fun  f }  =  {
f  e.  ~P (
y  X.  A )  |  Fun  f } )
1311, 12syl 14 . . . . 5  |-  ( x  =  A  ->  { f  e.  ~P ( y  X.  x )  |  Fun  f }  =  { f  e.  ~P ( y  X.  A
)  |  Fun  f } )
14 xpeq1 4521 . . . . . . 7  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
1514pweqd 3483 . . . . . 6  |-  ( y  =  B  ->  ~P ( y  X.  A
)  =  ~P ( B  X.  A ) )
16 rabeq 2650 . . . . . 6  |-  ( ~P ( y  X.  A
)  =  ~P ( B  X.  A )  ->  { f  e.  ~P ( y  X.  A
)  |  Fun  f }  =  { f  e.  ~P ( B  X.  A )  |  Fun  f } )
1715, 16syl 14 . . . . 5  |-  ( y  =  B  ->  { f  e.  ~P ( y  X.  A )  |  Fun  f }  =  { f  e.  ~P ( B  X.  A
)  |  Fun  f } )
18 df-pm 6511 . . . . 5  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
1913, 17, 18ovmpog 5871 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  {
f  e.  ~P ( B  X.  A )  |  Fun  f }  e.  _V )  ->  ( A 
^pm  B )  =  { f  e.  ~P ( B  X.  A
)  |  Fun  f } )
20193expia 1166 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { f  e. 
~P ( B  X.  A )  |  Fun  f }  e.  _V  ->  ( A  ^pm  B
)  =  { f  e.  ~P ( B  X.  A )  |  Fun  f } ) )
218, 9, 20syl2an 285 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { f  e. 
~P ( B  X.  A )  |  Fun  f }  e.  _V  ->  ( A  ^pm  B
)  =  { f  e.  ~P ( B  X.  A )  |  Fun  f } ) )
227, 21mpd 13 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  ^pm  B
)  =  { f  e.  ~P ( B  X.  A )  |  Fun  f } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   {crab 2395   _Vcvv 2658    C_ wss 3039   ~Pcpw 3478    X. cxp 4505   Fun wfun 5085  (class class class)co 5740    ^pm cpm 6509
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-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pm 6511
This theorem is referenced by:  elpmg  6524
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