Theorem fvresex 7963

Description: Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
fvresex.1	⊢ 𝐴 ∈ V

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fvresex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3988	. . . . . . . 8 ⊢ 𝐴 ⊆ V
2		resmpt 6029	. . . . . . . 8 ⊢ (𝐴 ⊆ V → ((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴) = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ ((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴) = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))
4	3	fveq1i 6882	. . . . . 6 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)
5		fveq2 6881	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
6		eqid 2736	. . . . . . . . 9 ⊢ (𝑧 ∈ V ↦ (𝐹‘𝑧)) = (𝑧 ∈ V ↦ (𝐹‘𝑧))
7		fvex 6894	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
8	5, 6, 7	fvmpt 6991	. . . . . . . 8 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥))
9	8	elv 3469	. . . . . . 7 ⊢ ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥)
10		fveqres 6928	. . . . . . 7 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥) → (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥))
11	9, 10	ax-mp 5	. . . . . 6 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥)
12	4, 11	eqtr3i 2761	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥)
13	12	eqeq2i 2749	. . . 4 ⊢ (𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) ↔ 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥))
14	13	exbii 1848	. . 3 ⊢ (∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) ↔ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥))
15	14	abbii 2803	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)} = {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)}
16		fvresex.1	. . . 4 ⊢ 𝐴 ∈ V
17	16	mptex 7220	. . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ∈ V
18	17	fvclex 7962	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)} ∈ V
19	15, 18	eqeltrri 2832	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)} ∈ V

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2714  Vcvv 3464   ⊆ wss 3931   ↦ cmpt 5206   ↾ cres 5661  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by: (None)