Theorem elpm 5395

Description: The predicate "is a partial function."

Hypotheses

Ref	Expression
elmap.1	|- A e. _V
elmap.2	|- B e. _V

Assertion

Ref	Expression
elpm	|- (F e. (A ^pm B) <-> (Fun F /\ F C_ (B X. A)))

Proof of Theorem elpm

Step	Hyp	Ref	Expression
1		elmap.1	. . . 4 |- A e. _V
2		elmap.2	. . . 4 |- B e. _V
3		pmvalg 5390	. . . 4 |- ((A e. _V /\ B e. _V) -> (A ^pm B) = {g | (Fun g /\ g C_ (B X. A))})
4	1, 2, 3	mp2an 761	. . 3 |- (A ^pm B) = {g | (Fun g /\ g C_ (B X. A))}
5	4	eleq2i 1961	. 2 |- (F e. (A ^pm B) <-> F e. {g | (Fun g /\ g C_ (B X. A))})
6	2, 1	xpex 4096	. . . . 5 |- (B X. A) e. _V
7	6	ssex 3455	. . . 4 |- (F C_ (B X. A) -> F e. _V)
8	7	adantl 424	. . 3 |- ((Fun F /\ F C_ (B X. A)) -> F e. _V)
9		funeq 4441	. . . 4 |- (g = F -> (Fun g <-> Fun F))
10		sseq1 2637	. . . 4 |- (g = F -> (g C_ (B X. A) <-> F C_ (B X. A)))
11	9, 10	anbi12d 690	. . 3 |- (g = F -> ((Fun g /\ g C_ (B X. A)) <-> (Fun F /\ F C_ (B X. A))))
12	8, 11	elab3 2412	. 2 |- (F e. {g | (Fun g /\ g C_ (B X. A))} <-> (Fun F /\ F C_ (B X. A)))
13	5, 12	bitri 190	1 |- (F e. (A ^pm B) <-> (Fun F /\ F C_ (B X. A)))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   C_ wss 2593   X. cxp 3984  Fun wfun 3992  (class class class)co 4884   ^pm cpm 5382

This theorem is referenced by:  elpm2 5396

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-pm 5384

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