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Theorem fvresex 7897
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 3973 . . . . . . . 8 𝐴 ⊆ V
2 resmpt 5996 . . . . . . . 8 (𝐴 ⊆ V → ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧)))
31, 2ax-mp 5 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧))
43fveq1i 6848 . . . . . 6 (((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)
5 fveq2 6847 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
6 eqid 2737 . . . . . . . . 9 (𝑧 ∈ V ↦ (𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐹𝑧))
7 fvex 6860 . . . . . . . . 9 (𝐹𝑥) ∈ V
85, 6, 7fvmpt 6953 . . . . . . . 8 (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥))
98elv 3454 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥)
10 fveqres 6894 . . . . . . 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 2767 . . . . 5 ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) = ((𝐹𝐴)‘𝑥)
1312eqeq2i 2750 . . . 4 (𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) ↔ 𝑦 = ((𝐹𝐴)‘𝑥))
1413exbii 1851 . . 3 (∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) ↔ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥))
1514abbii 2807 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} = {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)}
16 fvresex.1 . . . 4 𝐴 ∈ V
1716mptex 7178 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) ∈ V
1817fvclex 7896 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} ∈ V
1915, 18eqeltrri 2835 1 {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  wcel 2107  {cab 2714  Vcvv 3448  wss 3915  cmpt 5193  cres 5640  cfv 6501
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509
This theorem is referenced by: (None)
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