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Theorem pmvalg 7468
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 3524 . . 3  |-  { f  e.  ~P ( B  X.  A )  |  Fun  f }  C_  ~P ( B  X.  A
)
2 xpexg 6584 . . . . 5  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( B  X.  A
)  e.  _V )
32ancoms 451 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( B  X.  A
)  e.  _V )
4 pwexg 4578 . . . 4  |-  ( ( B  X.  A )  e.  _V  ->  ~P ( B  X.  A
)  e.  _V )
53, 4syl 17 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ~P ( B  X.  A )  e.  _V )
6 ssexg 4540 . . 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 661 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  e.  ~P ( B  X.  A
)  |  Fun  f }  e.  _V )
8 elex 3068 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
9 elex 3068 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
10 xpeq2 4838 . . . . . . 7  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
1110pweqd 3960 . . . . . 6  |-  ( x  =  A  ->  ~P ( y  X.  x
)  =  ~P (
y  X.  A ) )
12 rabeq 3053 . . . . . 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 17 . . . . 5  |-  ( x  =  A  ->  { f  e.  ~P ( y  X.  x )  |  Fun  f }  =  { f  e.  ~P ( y  X.  A
)  |  Fun  f } )
14 xpeq1 4837 . . . . . . 7  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
1514pweqd 3960 . . . . . 6  |-  ( y  =  B  ->  ~P ( y  X.  A
)  =  ~P ( B  X.  A ) )
16 rabeq 3053 . . . . . 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 17 . . . . 5  |-  ( y  =  B  ->  { f  e.  ~P ( y  X.  A )  |  Fun  f }  =  { f  e.  ~P ( B  X.  A
)  |  Fun  f } )
18 df-pm 7460 . . . . 5  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
1913, 17, 18ovmpt2g 6418 . . . 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 1199 . . 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 475 . 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 15 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  ^pm  B
)  =  { f  e.  ~P ( B  X.  A )  |  Fun  f } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {crab 2758   _Vcvv 3059    C_ wss 3414   ~Pcpw 3955    X. cxp 4821   Fun wfun 5563  (class class class)co 6278    ^pm cpm 7458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-pm 7460
This theorem is referenced by:  elpmg  7472
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