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Theorem fvresex 7656
 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 = ((𝐹𝐴)‘𝑥)} ∈ V
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fvresex
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ssv 3941 . . . . . . . 8 𝐴 ⊆ V
2 resmpt 5876 . . . . . . . 8 (𝐴 ⊆ V → ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧)))
31, 2ax-mp 5 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧))
43fveq1i 6656 . . . . . 6 (((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)
5 fveq2 6655 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
6 eqid 2798 . . . . . . . . 9 (𝑧 ∈ V ↦ (𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐹𝑧))
7 fvex 6668 . . . . . . . . 9 (𝐹𝑥) ∈ V
85, 6, 7fvmpt 6755 . . . . . . . 8 (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥))
98elv 3447 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥)
10 fveqres 6697 . . . . . . 7 (((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥) → (((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹𝐴)‘𝑥))
119, 10ax-mp 5 . . . . . 6 (((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹𝐴)‘𝑥)
124, 11eqtr3i 2823 . . . . 5 ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) = ((𝐹𝐴)‘𝑥)
1312eqeq2i 2811 . . . 4 (𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) ↔ 𝑦 = ((𝐹𝐴)‘𝑥))
1413exbii 1849 . . 3 (∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) ↔ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥))
1514abbii 2863 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} = {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)}
16 fvresex.1 . . . 4 𝐴 ∈ V
1716mptex 6973 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) ∈ V
1817fvclex 7655 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} ∈ V
1915, 18eqeltrri 2887 1 {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)} ∈ V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  Vcvv 3442   ⊆ wss 3883   ↦ cmpt 5114   ↾ cres 5525  ‘cfv 6332 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340 This theorem is referenced by: (None)
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