Theorem elpmg 6751

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 6746	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( A ^pm B ) = { g e. ~P ( B X. A ) | Fun g } )
2	1	eleq2d 2275	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^pm B ) <-> C e. { g e. ~P ( B X. A ) | Fun g } ) )
3		funeq 5291	. . . . 5 |- ( g = C -> ( Fun g <-> Fun C ) )
4	3	elrab 2929	. . . 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 2783	. . . . 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 4789	. . . . . 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 4183	. . . . . 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 3624	. . . . 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 707	. . 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 2176   {crab 2488   _Vcvv 2772   C_ wss 3166   ~Pcpw 3616   X. cxp 4673   Fun wfun 5265  (class class class)co 5944   ^pm cpm 6736

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pm 6738

This theorem is referenced by:  elpm2g  6752  pmss12g  6762  elpm  6766  pmsspw  6770  ennnfonelemj0  12772  lmfss  14716