MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pmvalg Unicode version

Theorem pmvalg 6799
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 3271 . . 3  |-  { f  e.  ~P ( B  X.  A )  |  Fun  f }  C_  ~P ( B  X.  A
)
2 xpexg 4816 . . . . 5  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( B  X.  A
)  e.  _V )
32ancoms 439 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( B  X.  A
)  e.  _V )
4 pwexg 4210 . . . 4  |-  ( ( B  X.  A )  e.  _V  ->  ~P ( B  X.  A
)  e.  _V )
53, 4syl 15 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ~P ( B  X.  A )  e.  _V )
6 ssexg 4176 . . 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 644 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  e.  ~P ( B  X.  A
)  |  Fun  f }  e.  _V )
8 elex 2809 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
9 elex 2809 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
10 xpeq2 4720 . . . . . . 7  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
1110pweqd 3643 . . . . . 6  |-  ( x  =  A  ->  ~P ( y  X.  x
)  =  ~P (
y  X.  A ) )
12 rabeq 2795 . . . . . 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 15 . . . . 5  |-  ( x  =  A  ->  { f  e.  ~P ( y  X.  x )  |  Fun  f }  =  { f  e.  ~P ( y  X.  A
)  |  Fun  f } )
14 xpeq1 4719 . . . . . . 7  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
1514pweqd 3643 . . . . . 6  |-  ( y  =  B  ->  ~P ( y  X.  A
)  =  ~P ( B  X.  A ) )
16 rabeq 2795 . . . . . 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 15 . . . . 5  |-  ( y  =  B  ->  { f  e.  ~P ( y  X.  A )  |  Fun  f }  =  { f  e.  ~P ( B  X.  A
)  |  Fun  f } )
18 df-pm 6791 . . . . 5  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
1913, 17, 18ovmpt2g 5998 . . . 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 1153 . . 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 463 . 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 14 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 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638    X. cxp 4703   Fun wfun 5265  (class class class)co 5874    ^pm cpm 6789
This theorem is referenced by:  elpmg  6802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791
  Copyright terms: Public domain W3C validator