Theorem fvresex 7660

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 3990	. . . . . . . 8 ⊢ 𝐴 ⊆ V
2		resmpt 5904	. . . . . . . 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 6670	. . . . . 6 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)
5		fveq2 6669	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
6		eqid 2821	. . . . . . . . 9 ⊢ (𝑧 ∈ V ↦ (𝐹‘𝑧)) = (𝑧 ∈ V ↦ (𝐹‘𝑧))
7		fvex 6682	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
8	5, 6, 7	fvmpt 6767	. . . . . . . 8 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥))
9	8	elv 3499	. . . . . . 7 ⊢ ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥)
10		fveqres 6711	. . . . . . 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 2846	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥)
13	12	eqeq2i 2834	. . . 4 ⊢ (𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) ↔ 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥))
14	13	exbii 1844	. . 3 ⊢ (∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) ↔ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥))
15	14	abbii 2886	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)} = {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)}
16		fvresex.1	. . . 4 ⊢ 𝐴 ∈ V
17	16	mptex 6985	. . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ∈ V
18	17	fvclex 7659	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)} ∈ V
19	15, 18	eqeltrri 2910	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)} ∈ V

Syntax hints:   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799  Vcvv 3494   ⊆ wss 3935   ↦ cmpt 5145   ↾ cres 5556  ‘cfv 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362

This theorem is referenced by: (None)