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Theorem fvresex 7969
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 4006 . . . . . . . 8 𝐴 ⊆ V
2 resmpt 6046 . . . . . . . 8 (𝐴 ⊆ V → ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧)))
31, 2ax-mp 5 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧))
43fveq1i 6903 . . . . . 6 (((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)
5 fveq2 6902 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
6 eqid 2728 . . . . . . . . 9 (𝑧 ∈ V ↦ (𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐹𝑧))
7 fvex 6915 . . . . . . . . 9 (𝐹𝑥) ∈ V
85, 6, 7fvmpt 7010 . . . . . . . 8 (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥))
98elv 3479 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥)
10 fveqres 6949 . . . . . . 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 2758 . . . . 5 ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) = ((𝐹𝐴)‘𝑥)
1312eqeq2i 2741 . . . 4 (𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) ↔ 𝑦 = ((𝐹𝐴)‘𝑥))
1413exbii 1842 . . 3 (∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) ↔ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥))
1514abbii 2798 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} = {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)}
16 fvresex.1 . . . 4 𝐴 ∈ V
1716mptex 7241 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) ∈ V
1817fvclex 7968 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} ∈ V
1915, 18eqeltrri 2826 1 {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wex 1773  wcel 2098  {cab 2705  Vcvv 3473  wss 3949  cmpt 5235  cres 5684  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561
This theorem is referenced by: (None)
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