Theorem pmsspw 6661

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 6629	. . . . . . 7 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2	1	elmpocl 6047	. . . . . 6 |- ( f e. ( A ^pm B ) -> ( A e. _V /\ B e. _V ) )
3		elpmg 6642	. . . . . 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 3573	. . 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 3151	1 |- ( A ^pm B ) C_ ~P ( B X. A )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 2141   {crab 2452   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566   X. cxp 4609   Fun wfun 5192  (class class class)co 5853   ^pm cpm 6627

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pm 6629

This theorem is referenced by:  mapsspw  6662