Theorem pmsspw 6649

Description: Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
pmsspw	|- ( A ^pm B ) C_ ~P ( B X. A )

Proof of Theorem pmsspw

Dummy variables f x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pm 6617	. . . . . . 7 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2	1	elmpocl 6036	. . . . . 6 |- ( f e. ( A ^pm B ) -> ( A e. _V /\ B e. _V ) )
3		elpmg 6630	. . . . . 6 |- ( ( A e. _V /\ B e. _V ) -> ( f e. ( A ^pm B ) <-> ( Fun f /\ f C_ ( B X. A ) ) ) )
4	2, 3	syl 14	. . . . 5 |- ( f e. ( A ^pm B ) -> ( f e. ( A ^pm B ) <-> ( Fun f /\ f C_ ( B X. A ) ) ) )
5	4	ibi 175	. . . 4 |- ( f e. ( A ^pm B ) -> ( Fun f /\ f C_ ( B X. A ) ) )
6	5	simprd 113	. . 3 |- ( f e. ( A ^pm B ) -> f C_ ( B X. A ) )
7		velpw 3566	. . 3 |- ( f e. ~P ( B X. A ) <-> f C_ ( B X. A ) )
8	6, 7	sylibr 133	. 2 |- ( f e. ( A ^pm B ) -> f e. ~P ( B X. A ) )
9	8	ssriv 3146	1 |- ( A ^pm B ) C_ ~P ( B X. A )

Syntax hints:   /\ wa 103   <-> wb 104   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:  mapsspw  6650