Theorem elpmg 6014

Description: The predicate "is a partial function." (Contributed by set.mm contributors, 14-Nov-2013.)

Assertion

Ref	Expression
elpmg	|- ((A e. V /\ B e. W /\ C e. X) -> (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 6011	. . . . 5 |- ((A e. V /\ B e. W) -> (A ^pm B) = {g e. ~P(B X. A) | Fun g})
2	1	eleq2d 2420	. . . 4 |- ((A e. V /\ B e. W) -> (C e. (A ^pm B) <-> C e. {g e. ~P(B X. A) | Fun g}))
3		funeq 5128	. . . . 5 |- (g = C -> (Fun g <-> Fun C))
4	3	elrab 2995	. . . 4 |- (C e. {g e. ~P(B X. A) | Fun g} <-> (C e. ~P(B X. A) /\ Fun C))
5	2, 4	syl6bb 252	. . 3 |- ((A e. V /\ B e. W) -> (C e. (A ^pm B) <-> (C e. ~P(B X. A) /\ Fun C)))
6	5	3adant3 975	. 2 |- ((A e. V /\ B e. W /\ C e. X) -> (C e. (A ^pm B) <-> (C e. ~P(B X. A) /\ Fun C)))
7		elpwg 3730	. . . . 5 |- (C e. X -> (C e. ~P(B X. A) <-> C C_ (B X. A)))
8	7	anbi1d 685	. . . 4 |- (C e. X -> ((C e. ~P(B X. A) /\ Fun C) <-> (C C_ (B X. A) /\ Fun C)))
9		ancom 437	. . . 4 |- ((C C_ (B X. A) /\ Fun C) <-> (Fun C /\ C C_ (B X. A)))
10	8, 9	syl6bb 252	. . 3 |- (C e. X -> ((C e. ~P(B X. A) /\ Fun C) <-> (Fun C /\ C C_ (B X. A))))
11	10	3ad2ant3 978	. 2 |- ((A e. V /\ B e. W /\ C e. X) -> ((C e. ~P(B X. A) /\ Fun C) <-> (Fun C /\ C C_ (B X. A))))
12	6, 11	bitrd 244	1 |- ((A e. V /\ B e. W /\ C e. X) -> (C e. (A ^pm B) <-> (Fun C /\ C C_ (B X. A))))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   e. wcel 1710  {crab 2619   C_ wss 3258  ~Pcpw 3723   X. cxp 4771  Fun wfun 4776  (class class class)co 5526   ^pm cpm 6001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759  df-funs 5761  df-pm 6003

This theorem is referenced by:  elpm2g  6015  pmfun  6016  elpm  6020