Theorem elpmg 6723

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 6718	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( A ^pm B ) = { g e. ~P ( B X. A ) | Fun g } )
2	1	eleq2d 2266	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> C e. { g e. ~P ( B X. A ) | Fun g } ) )
3		funeq 5278	. . . . 5 |- ( g = C -> ( Fun g <-> Fun C ) )
4	3	elrab 2920	. . . 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 2774	. . . . 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 4777	. . . . . 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 4172	. . . . . 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 3613	. . . . 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 2167   {crab 2479   _Vcvv 2763   C_ wss 3157   ~Pcpw 3605   X. cxp 4661   Fun wfun 5252  (class class class)co 5922   ^pm cpm 6708

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pm 6710

This theorem is referenced by:  elpm2g  6724  pmss12g  6734  elpm  6738  pmsspw  6742  ennnfonelemj0  12618  lmfss  14480