Theorem pmsspw 6480

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 6448	. . . . . . 7 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2	1	elmpt2cl 5880	. . . . . 6 |- ( f e. ( A ^pm B ) -> ( A e. _V /\ B e. _V ) )
3		elpmg 6461	. . . . . 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		selpw 3456	. . 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 3043	1 |- ( A ^pm B ) C_ ~P ( B X. A )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 1445   {crab 2374   _Vcvv 2633   C_ wss 3013   ~Pcpw 3449   X. cxp 4465   Fun wfun 5043  (class class class)co 5690   ^pm cpm 6446

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pm 6448

This theorem is referenced by:  mapsspw  6481