Theorem fvresex 2909

Description: Existence of the class of values of a restricted class.

Hypothesis

Ref	Expression
fvresex.1	⊢ A ∈ V

Assertion

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

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

Proof of Theorem fvresex

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . 7 ⊢ x ∈ V
2		fvex 2838	. . . . . . 7 ⊢ (F ‘x) ∈ V
3		fveq2 2832	. . . . . . 7 ⊢ (z = x → (F ‘z) = (F ‘x))
4	1, 2, 3	fvopab 2877	. . . . . 6 ⊢ ({⟨z, w⟩∣w = (F ‘z)} ‘x) = (F ‘x)
5		fveqres 2851	. . . . . 6 ⊢ (({⟨z, w⟩∣w = (F ‘z)} ‘x) = (F ‘x) → (({⟨z, w⟩∣w = (F ‘z)} ↾ A) ‘x) = ((F ↾ A) ‘x))
6	4, 5	ax-mp 6	. . . . 5 ⊢ (({⟨z, w⟩∣w = (F ‘z)} ↾ A) ‘x) = ((F ↾ A) ‘x)
7	6	cleq2i 1111	. . . 4 ⊢ (y = (({⟨z, w⟩∣w = (F ‘z)} ↾ A) ‘x) ↔ y = ((F ↾ A) ‘x))
8	7	biex 733	. . 3 ⊢ (∃x y = (({⟨z, w⟩∣w = (F ‘z)} ↾ A) ‘x) ↔ ∃x y = ((F ↾ A) ‘x))
9	8	biabi 1181	. 2 ⊢ {y∣∃x y = (({⟨z, w⟩∣w = (F ‘z)} ↾ A) ‘x)} = {y∣∃x y = ((F ↾ A) ‘x)}
10		fvresex.1	. . . 4 ⊢ A ∈ V
11		funopabeq 2695	. . . 4 ⊢ Fun {⟨z, w⟩∣w = (F ‘z)}
12		resfunexg 2717	. . . 4 ⊢ (A ∈ V → (Fun {⟨z, w⟩∣w = (F ‘z)} → ({⟨z, w⟩∣w = (F ‘z)} ↾ A) ∈ V))
13	10, 11, 12	mp2 43	. . 3 ⊢ ({⟨z, w⟩∣w = (F ‘z)} ↾ A) ∈ V
14	13	fvclex 2908	. 2 ⊢ {y∣∃x y = (({⟨z, w⟩∣w = (F ‘z)} ↾ A) ‘x)} ∈ V
15	9, 14	eqeltrr 1160	1 ⊢ {y∣∃x y = ((F ↾ A) ‘x)} ∈ V

Syntax hints:  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {copab 2055   ↾ cres 2412  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  abrexexlem1 2910

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