Theorem pmsspw 6713

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 6681	. . . . . . 7 |- ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
2	1	elmpocl 6095	. . . . . 6 |- ( f e. ( A ^pm B ) -> ( A e. _V /\ B e. _V ) )
3		elpmg 6694	. . . . . 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 3600	. . 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 3174	1 |- ( A ^pm B ) C_ ~P ( B X. A )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2160   {crab 2472   _Vcvv 2752   C_ wss 3144   ~Pcpw 3593   X. cxp 4645   Fun wfun 5232  (class class class)co 5900   ^pm cpm 6679

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-pm 6681

This theorem is referenced by:  mapsspw  6714