Theorem elpm 4342

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 (_ (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 4337	. . . 4 |- ((A e. V /\ B e. V) -> (A ^pm B) = {g | (Fun g /\ g (_ (B X. A))})
4	1, 2, 3	mp2an 699	. . 3 |- (A ^pm B) = {g | (Fun g /\ g (_ (B X. A))}
5	4	eleq2i 1541	. 2 |- (F e. (A ^pm B) <-> F e. {g | (Fun g /\ g (_ (B X. A))})
6	2, 1	xpex 3266	. . . . 5 |- (B X. A) e. V
7	6	ssex 2724	. . . 4 |- (F (_ (B X. A) -> F e. V)
8	7	adantl 390	. . 3 |- ((Fun F /\ F (_ (B X. A)) -> F e. V)
9		funeq 3541	. . . 4 |- (g = F -> (Fun g <-> Fun F))
10		sseq1 2085	. . . 4 |- (g = F -> (g (_ (B X. A) <-> F (_ (B X. A)))
11	9, 10	anbi12d 630	. . 3 |- (g = F -> ((Fun g /\ g (_ (B X. A)) <-> (Fun F /\ F (_ (B X. A))))
12	8, 11	elab3 1906	. 2 |- (F e. {g | (Fun g /\ g (_ (B X. A))} <-> (Fun F /\ F (_ (B X. A)))
13	5, 12	bitr 173	1 |- (F e. (A ^pm B) <-> (Fun F /\ F (_ (B X. A)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  Vcvv 1814   (_ wss 2050   X. cxp 3174  Fun wfun 3182  (class class class)co 3969   ^pm cpm 4329

This theorem is referenced by:  elpm2 4343

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-opr 3971  df-oprab 3972  df-pm 4331

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