Theorem elpmg 6832

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 6827	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( A ^pm B ) = { g e. ~P ( B X. A ) | Fun g } )
2	1	eleq2d 2301	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> C e. { g e. ~P ( B X. A ) | Fun g } ) )
3		funeq 5346	. . . . 5 |- ( g = C -> ( Fun g <-> Fun C ) )
4	3	elrab 2962	. . . 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 2814	. . . . 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 4840	. . . . . 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 4228	. . . . . 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 3660	. . . . 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 712	. . 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 2202   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   X. cxp 4723   Fun wfun 5320  (class class class)co 6017   ^pm cpm 6817

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pm 6819

This theorem is referenced by:  elpm2g  6833  pmss12g  6843  elpm  6847  pmsspw  6851  ennnfonelemj0  13021  lmfss  14967