Theorem pmvalg 6625

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 3227	. . 3 |- { f e. ~P ( B X. A ) | Fun f } C_ ~P ( B X. A )
2		xpexg 4718	. . . . 5 |- ( ( B e. D /\ A e. C ) -> ( B X. A ) e. _V )
3	2	ancoms 266	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( B X. A ) e. _V )
4		pwexg 4159	. . . 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 4121	. . 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 411	. 2 |- ( ( A e. C /\ B e. D ) -> { f e. ~P ( B X. A ) | Fun f } e. _V )
8		elex 2737	. . 3 |- ( A e. C -> A e. _V )
9		elex 2737	. . 3 |- ( B e. D -> B e. _V )
10		xpeq2 4619	. . . . . . 7 |- ( x = A -> ( y X. x ) = ( y X. A ) )
11	10	pweqd 3564	. . . . . 6 |- ( x = A -> ~P ( y X. x ) = ~P ( y X. A ) )
12		rabeq 2718	. . . . . 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 4618	. . . . . . 7 |- ( y = B -> ( y X. A ) = ( B X. A ) )
15	14	pweqd 3564	. . . . . 6 |- ( y = B -> ~P ( y X. A ) = ~P ( B X. A ) )
16		rabeq 2718	. . . . . 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 6617	. . . . 5 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
19	13, 17, 18	ovmpog 5976	. . . 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 1195	. . 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 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 } ) )
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 103   = wceq 1343   e. wcel 2136   {crab 2448   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   X. cxp 4602   Fun wfun 5182  (class class class)co 5842   ^pm cpm 6615

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pm 6617

This theorem is referenced by:  elpmg  6630