Theorem pmvalg 6923

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 3333	. . 3 |- { f e. ~P ( B X. A ) | Fun f } C_ ~P ( B X. A )
2		xpexg 4884	. . . . 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 4312	. . . 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 4267	. . 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 418	. 2 |- ( ( A e. C /\ B e. D ) -> { f e. ~P ( B X. A ) | Fun f } e. _V )
8		elex 2833	. . 3 |- ( A e. C -> A e. _V )
9		elex 2833	. . 3 |- ( B e. D -> B e. _V )
10		xpeq2 4784	. . . . . . 7 |- ( x = A -> ( y X. x ) = ( y X. A ) )
11	10	pweqd 3690	. . . . . 6 |- ( x = A -> ~P ( y X. x ) = ~P ( y X. A ) )
12		rabeq 2813	. . . . . 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 4783	. . . . . . 7 |- ( y = B -> ( y X. A ) = ( B X. A ) )
15	14	pweqd 3690	. . . . . 6 |- ( y = B -> ~P ( y X. A ) = ~P ( B X. A ) )
16		rabeq 2813	. . . . . 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 6915	. . . . 5 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
19	13, 17, 18	ovmpog 6213	. . . 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 1236	. . 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 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685   X. cxp 4767   Fun wfun 5366  (class class class)co 6075   ^pm cpm 6913

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pm 6915

This theorem is referenced by:  elpmg  6928