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Theorem pmvalg 6637
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 3232 . . 3  |-  { f  e.  ~P ( B  X.  A )  |  Fun  f }  C_  ~P ( B  X.  A
)
2 xpexg 4725 . . . . 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 4166 . . . 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 4128 . . 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 412 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  e.  ~P ( B  X.  A
)  |  Fun  f }  e.  _V )
8 elex 2741 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
9 elex 2741 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
10 xpeq2 4626 . . . . . . 7  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
1110pweqd 3571 . . . . . 6  |-  ( x  =  A  ->  ~P ( y  X.  x
)  =  ~P (
y  X.  A ) )
12 rabeq 2722 . . . . . 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 4625 . . . . . . 7  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
1514pweqd 3571 . . . . . 6  |-  ( y  =  B  ->  ~P ( y  X.  A
)  =  ~P ( B  X.  A ) )
16 rabeq 2722 . . . . . 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 6629 . . . . 5  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
1913, 17, 18ovmpog 5987 . . . 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 1200 . . 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 287 . 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 1348    e. wcel 2141   {crab 2452   _Vcvv 2730    C_ wss 3121   ~Pcpw 3566    X. cxp 4609   Fun wfun 5192  (class class class)co 5853    ^pm cpm 6627
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pm 6629
This theorem is referenced by:  elpmg  6642
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