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

Theorem pmvalg 7246
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 3458 . . 3  |-  { f  e.  ~P ( B  X.  A )  |  Fun  f }  C_  ~P ( B  X.  A
)
2 xpexg 6528 . . . . 5  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( B  X.  A
)  e.  _V )
32ancoms 453 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( B  X.  A
)  e.  _V )
4 pwexg 4497 . . . 4  |-  ( ( B  X.  A )  e.  _V  ->  ~P ( B  X.  A
)  e.  _V )
53, 4syl 16 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ~P ( B  X.  A )  e.  _V )
6 ssexg 4459 . . 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 663 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  e.  ~P ( B  X.  A
)  |  Fun  f }  e.  _V )
8 elex 3002 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
9 elex 3002 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
10 xpeq2 4876 . . . . . . 7  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
1110pweqd 3886 . . . . . 6  |-  ( x  =  A  ->  ~P ( y  X.  x
)  =  ~P (
y  X.  A ) )
12 rabeq 2987 . . . . . 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 16 . . . . 5  |-  ( x  =  A  ->  { f  e.  ~P ( y  X.  x )  |  Fun  f }  =  { f  e.  ~P ( y  X.  A
)  |  Fun  f } )
14 xpeq1 4875 . . . . . . 7  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
1514pweqd 3886 . . . . . 6  |-  ( y  =  B  ->  ~P ( y  X.  A
)  =  ~P ( B  X.  A ) )
16 rabeq 2987 . . . . . 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 16 . . . . 5  |-  ( y  =  B  ->  { f  e.  ~P ( y  X.  A )  |  Fun  f }  =  { f  e.  ~P ( B  X.  A
)  |  Fun  f } )
18 df-pm 7238 . . . . 5  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
1913, 17, 18ovmpt2g 6246 . . . 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 1189 . . 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 477 . 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 369    = wceq 1369    e. wcel 1756   {crab 2740   _Vcvv 2993    C_ wss 3349   ~Pcpw 3881    X. cxp 4859   Fun wfun 5433  (class class class)co 6112    ^pm cpm 7236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-pm 7238
This theorem is referenced by:  elpmg  7249
  Copyright terms: Public domain W3C validator