Theorem pmvalg 6713

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 3264	. . 3 |- { f e. ~P ( B X. A ) | Fun f } C_ ~P ( B X. A )
2		xpexg 4773	. . . . 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 4209	. . . 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 4168	. . 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 2771	. . 3 |- ( A e. C -> A e. _V )
9		elex 2771	. . 3 |- ( B e. D -> B e. _V )
10		xpeq2 4674	. . . . . . 7 |- ( x = A -> ( y X. x ) = ( y X. A ) )
11	10	pweqd 3606	. . . . . 6 |- ( x = A -> ~P ( y X. x ) = ~P ( y X. A ) )
12		rabeq 2752	. . . . . 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 4673	. . . . . . 7 |- ( y = B -> ( y X. A ) = ( B X. A ) )
15	14	pweqd 3606	. . . . . 6 |- ( y = B -> ~P ( y X. A ) = ~P ( B X. A ) )
16		rabeq 2752	. . . . . 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 6705	. . . . 5 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
19	13, 17, 18	ovmpog 6053	. . . 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 1364   e. wcel 2164   {crab 2476   _Vcvv 2760   C_ wss 3153   ~Pcpw 3601   X. cxp 4657   Fun wfun 5248  (class class class)co 5918   ^pm cpm 6703

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pm 6705

This theorem is referenced by:  elpmg  6718