Theorem fvresex 7946

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 4007	. . . . . . . 8 ⊢ 𝐴 ⊆ V
2		resmpt 6038	. . . . . . . 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 6893	. . . . . 6 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)
5		fveq2 6892	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
6		eqid 2733	. . . . . . . . 9 ⊢ (𝑧 ∈ V ↦ (𝐹‘𝑧)) = (𝑧 ∈ V ↦ (𝐹‘𝑧))
7		fvex 6905	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
8	5, 6, 7	fvmpt 6999	. . . . . . . 8 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥))
9	8	elv 3481	. . . . . . 7 ⊢ ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥)
10		fveqres 6939	. . . . . . 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 2763	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥)
13	12	eqeq2i 2746	. . . 4 ⊢ (𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) ↔ 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥))
14	13	exbii 1851	. . 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 7225	. . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ∈ V
18	17	fvclex 7945	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)} ∈ V
19	15, 18	eqeltrri 2831	1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)} ∈ V

Syntax hints:   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710  Vcvv 3475   ⊆ wss 3949   ↦ cmpt 5232   ↾ cres 5679  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552

This theorem is referenced by: (None)