Theorem abrexexlem1 2910

Description: Lemma for abrexex 2912. Shows the existence of a class of existentially restricted function values.

Hypothesis

Ref	Expression
abrexexlem1.1	|- A e. V

Assertion

Ref	Expression
abrexexlem1	|- {y | E.x e. A y = (F` x)} e. V

Distinct variable group(s):   x,y,F   x,A,y

Proof of Theorem abrexexlem1

Step	Hyp	Ref	Expression
1		abrexexlem1.1	. . 3 |- A e. V
2	1	fvresex 2909	. 2 |- {y | E.x y = ((F |` A)` x)} e. V
3		df-rex 1206	. . . 4 |- (E.x e. A y = (F` x) <-> E.x(x e. A /\ y = (F` x)))
4		fvres 2840	. . . . . . 7 |- (x e. A -> ((F |` A)` x) = (F` x))
5	4	cleq2d 1112	. . . . . 6 |- (x e. A -> (y = ((F |` A)` x) <-> y = (F` x)))
6	5	biimpar 325	. . . . 5 |- ((x e. A /\ y = (F` x)) -> y = ((F |` A)` x))
7	6	19.22i 723	. . . 4 |- (E.x(x e. A /\ y = (F` x)) -> E.x y = ((F |` A)` x))
8	3, 7	sylbi 174	. . 3 |- (E.x e. A y = (F` x) -> E.x y = ((F |` A)` x))
9	8	ss2abi 1552	. 2 |- {y | E.x e. A y = (F` x)} (_ {y | E.x y = ((F |` A)` x)}
10	2, 9	ssexi 1701	1 |- {y | E.x e. A y = (F` x)} e. V

Syntax hints:   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  E.wrex 1202  Vcvv 1348   |` cres 2412  ` cfv 2422

This theorem is referenced by:  abrexexlem2 2911

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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