Theorem pmvalg 6745

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.

Step	Hyp	Ref	Expression
1		ssrab2 3277	. . 3 |- { f e. ~P ( B X. A ) | Fun f } C_ ~P ( B X. A )
2		xpexg 4788	. . . . 5 |- ( ( B e. D /\ A e. C ) -> ( B X. A ) e. _V )
3	2	ancoms 268	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( B X. A ) e. _V )
4		pwexg 4223	. . . 4 |- ( ( B X. A ) e. _V -> ~P ( B X. A ) e. _V )
5	3, 4	syl 14	. . 3 |- ( ( A e. C /\ B e. D ) -> ~P ( B X. A ) e. _V )
6		ssexg 4182	. . 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 )
7	1, 5, 6	sylancr 414	. 2 |- ( ( A e. C /\ B e. D ) -> { f e. ~P ( B X. A ) | Fun f } e. _V )
8		elex 2782	. . 3 |- ( A e. C -> A e. _V )
9		elex 2782	. . 3 |- ( B e. D -> B e. _V )
10		xpeq2 4689	. . . . . . 7 |- ( x = A -> ( y X. x ) = ( y X. A ) )
11	10	pweqd 3620	. . . . . 6 |- ( x = A -> ~P ( y X. x ) = ~P ( y X. A ) )
12		rabeq 2763	. . . . . 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 } )
13	11, 12	syl 14	. . . . 5 |- ( x = A -> { f e. ~P ( y X. x ) | Fun f } = { f e. ~P ( y X. A ) | Fun f } )
14		xpeq1 4688	. . . . . . 7 |- ( y = B -> ( y X. A ) = ( B X. A ) )
15	14	pweqd 3620	. . . . . 6 |- ( y = B -> ~P ( y X. A ) = ~P ( B X. A ) )
16		rabeq 2763	. . . . . 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 } )
17	15, 16	syl 14	. . . . 5 |- ( y = B -> { f e. ~P ( y X. A ) | Fun f } = { f e. ~P ( B X. A ) | Fun f } )
18		df-pm 6737	. . . . 5 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
19	13, 17, 18	ovmpog 6079	. . . 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 } )
20	19	3expia 1207	. . 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 } ) )
21	8, 9, 20	syl2an 289	. 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 } ) )
22	7, 21	mpd 13	1 |- ( ( A e. C /\ B e. D ) -> ( A ^pm B ) = { f e. ~P ( B X. A ) | Fun f } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   {crab 2487   _Vcvv 2771   C_ wss 3165   ~Pcpw 3615   X. cxp 4672   Fun wfun 5264  (class class class)co 5943   ^pm cpm 6735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pm 6737

This theorem is referenced by:  elpmg  6750