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Theorem fvresex 7984
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 4008 . . . . . . . 8 𝐴 ⊆ V
2 resmpt 6055 . . . . . . . 8 (𝐴 ⊆ V → ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧)))
31, 2ax-mp 5 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧))
43fveq1i 6907 . . . . . 6 (((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)
5 fveq2 6906 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
6 eqid 2737 . . . . . . . . 9 (𝑧 ∈ V ↦ (𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐹𝑧))
7 fvex 6919 . . . . . . . . 9 (𝐹𝑥) ∈ V
85, 6, 7fvmpt 7016 . . . . . . . 8 (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥))
98elv 3485 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥)
10 fveqres 6953 . . . . . . 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 1848 . . 3 (∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) ↔ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥))
1514abbii 2809 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} = {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)}
16 fvresex.1 . . . 4 𝐴 ∈ V
1716mptex 7243 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) ∈ V
1817fvclex 7983 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} ∈ V
1915, 18eqeltrri 2838 1 {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wex 1779  wcel 2108  {cab 2714  Vcvv 3480  wss 3951  cmpt 5225  cres 5687  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by: (None)
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