Theorem pmsspw 6830

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 6798	. . . . . . 7 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2	1	elmpocl 6200	. . . . . 6 |- ( f e. ( A ^pm B ) -> ( A e. _V /\ B e. _V ) )
3		elpmg 6811	. . . . . 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 176	. . . 4 |- ( f e. ( A ^pm B ) -> ( Fun f /\ f C_ ( B X. A ) ) )
6	5	simprd 114	. . 3 |- ( f e. ( A ^pm B ) -> f C_ ( B X. A ) )
7		velpw 3656	. . 3 |- ( f e. ~P ( B X. A ) <-> f C_ ( B X. A ) )
8	6, 7	sylibr 134	. 2 |- ( f e. ( A ^pm B ) -> f e. ~P ( B X. A ) )
9	8	ssriv 3228	1 |- ( A ^pm B ) C_ ~P ( B X. A )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2200   {crab 2512   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   X. cxp 4717   Fun wfun 5312  (class class class)co 6001   ^pm cpm 6796

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pm 6798

This theorem is referenced by:  mapsspw  6831