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 ∈ V

Assertion

Ref	Expression
abrexexlem1	⊢ {y∣∃x ∈ A y = (F ‘x)} ∈ V

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

Proof of Theorem abrexexlem1

Step	Hyp	Ref	Expression
1		abrexexlem1.1	. . 3 ⊢ A ∈ V
2	1	fvresex 2909	. 2 ⊢ {y∣∃x y = ((F ↾ A) ‘x)} ∈ V
3		df-rex 1206	. . . 4 ⊢ (∃x ∈ A y = (F ‘x) ↔ ∃x(x ∈ A ∧ y = (F ‘x)))
4		fvres 2840	. . . . . . 7 ⊢ (x ∈ A → ((F ↾ A) ‘x) = (F ‘x))
5	4	cleq2d 1112	. . . . . 6 ⊢ (x ∈ A → (y = ((F ↾ A) ‘x) ↔ y = (F ‘x)))
6	5	biimpar 325	. . . . 5 ⊢ ((x ∈ A ∧ y = (F ‘x)) → y = ((F ↾ A) ‘x))
7	6	19.22i 723	. . . 4 ⊢ (∃x(x ∈ A ∧ y = (F ‘x)) → ∃x y = ((F ↾ A) ‘x))
8	3, 7	sylbi 174	. . 3 ⊢ (∃x ∈ A y = (F ‘x) → ∃x y = ((F ↾ A) ‘x))
9	8	ss2abi 1552	. 2 ⊢ {y∣∃x ∈ A y = (F ‘x)} ⊆ {y∣∃x y = ((F ↾ A) ‘x)}
10	2, 9	ssexi 1701	1 ⊢ {y∣∃x ∈ A y = (F ‘x)} ∈ V

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃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

metamath.org