Theorem elpmg 6718

Description: The predicate "is a partial function". (Contributed by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
elpmg	|- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> ( Fun C /\ C C_ ( B X. A ) ) ) )

Proof of Theorem elpmg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmvalg 6713	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( A ^pm B ) = { g e. ~P ( B X. A ) | Fun g } )
2	1	eleq2d 2263	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> C e. { g e. ~P ( B X. A ) | Fun g } ) )
3		funeq 5274	. . . . 5 |- ( g = C -> ( Fun g <-> Fun C ) )
4	3	elrab 2916	. . . 4 |- ( C e. { g e. ~P ( B X. A ) | Fun g } <-> ( C e. ~P ( B X. A ) /\ Fun C ) )
5	2, 4	bitrdi 196	. . 3 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> ( C e. ~P ( B X. A ) /\ Fun C ) ) )
6		ancom 266	. . 3 |- ( ( C e. ~P ( B X. A ) /\ Fun C ) <-> ( Fun C /\ C e. ~P ( B X. A ) ) )
7	5, 6	bitrdi 196	. 2 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> ( Fun C /\ C e. ~P ( B X. A ) ) ) )
8		elex 2771	. . . . 5 |- ( C e. ~P ( B X. A ) -> C e. _V )
9	8	a1i 9	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( C e. ~P ( B X. A ) -> C e. _V ) )
10		xpexg 4773	. . . . . 6 |- ( ( B e. W /\ A e. V ) -> ( B X. A ) e. _V )
11	10	ancoms 268	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( B X. A ) e. _V )
12		ssexg 4168	. . . . . 6 |- ( ( C C_ ( B X. A ) /\ ( B X. A ) e. _V ) -> C e. _V )
13	12	expcom 116	. . . . 5 |- ( ( B X. A ) e. _V -> ( C C_ ( B X. A ) -> C e. _V ) )
14	11, 13	syl 14	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( C C_ ( B X. A ) -> C e. _V ) )
15		elpwg 3609	. . . . 5 |- ( C e. _V -> ( C e. ~P ( B X. A ) <-> C C_ ( B X. A ) ) )
16	15	a1i 9	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( C e. _V -> ( C e. ~P ( B X. A ) <-> C C_ ( B X. A ) ) ) )
17	9, 14, 16	pm5.21ndd 706	. . 3 |- ( ( A e. V /\ B e. W ) -> ( C e. ~P ( B X. A ) <-> C C_ ( B X. A ) ) )
18	17	anbi2d 464	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( Fun C /\ C e. ~P ( B X. A ) ) <-> ( Fun C /\ C C_ ( B X. A ) ) ) )
19	7, 18	bitrd 188	1 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> ( Fun C /\ C C_ ( B X. A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2164   {crab 2476   _Vcvv 2760   C_ wss 3153   ~Pcpw 3601   X. cxp 4657   Fun wfun 5248  (class class class)co 5918   ^pm cpm 6703

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pm 6705

This theorem is referenced by:  elpm2g  6719  pmss12g  6729  elpm  6733  pmsspw  6737  ennnfonelemj0  12558  lmfss  14412