Theorem elpmg 6558

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 6553	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( A ^pm B ) = { g e. ~P ( B X. A ) | Fun g } )
2	1	eleq2d 2209	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> C e. { g e. ~P ( B X. A ) | Fun g } ) )
3		funeq 5143	. . . . 5 |- ( g = C -> ( Fun g <-> Fun C ) )
4	3	elrab 2840	. . . 4 |- ( C e. { g e. ~P ( B X. A ) | Fun g } <-> ( C e. ~P ( B X. A ) /\ Fun C ) )
5	2, 4	syl6bb 195	. . 3 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> ( C e. ~P ( B X. A ) /\ Fun C ) ) )
6		ancom 264	. . 3 |- ( ( C e. ~P ( B X. A ) /\ Fun C ) <-> ( Fun C /\ C e. ~P ( B X. A ) ) )
7	5, 6	syl6bb 195	. 2 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> ( Fun C /\ C e. ~P ( B X. A ) ) ) )
8		elex 2697	. . . . 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 4653	. . . . . 6 |- ( ( B e. W /\ A e. V ) -> ( B X. A ) e. _V )
11	10	ancoms 266	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( B X. A ) e. _V )
12		ssexg 4067	. . . . . 6 |- ( ( C C_ ( B X. A ) /\ ( B X. A ) e. _V ) -> C e. _V )
13	12	expcom 115	. . . . 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 3518	. . . . 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 694	. . 3 |- ( ( A e. V /\ B e. W ) -> ( C e. ~P ( B X. A ) <-> C C_ ( B X. A ) ) )
18	17	anbi2d 459	. 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 187	1 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> ( Fun C /\ C C_ ( B X. A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   {crab 2420   _Vcvv 2686   C_ wss 3071   ~Pcpw 3510   X. cxp 4537   Fun wfun 5117  (class class class)co 5774   ^pm cpm 6543

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pm 6545

This theorem is referenced by:  elpm2g  6559  pmss12g  6569  elpm  6573  pmsspw  6577  ennnfonelemj0  11914  lmfss  12413