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Theorem fvresex 7089
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 3606 . . . . . . . 8 𝐴 ⊆ V
2 resmpt 5410 . . . . . . . 8 (𝐴 ⊆ V → ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧)))
31, 2ax-mp 5 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴) = (𝑧𝐴 ↦ (𝐹𝑧))
43fveq1i 6151 . . . . . 6 (((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)
5 vex 3189 . . . . . . . 8 𝑥 ∈ V
6 fveq2 6150 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
7 eqid 2621 . . . . . . . . 9 (𝑧 ∈ V ↦ (𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐹𝑧))
8 fvex 6160 . . . . . . . . 9 (𝐹𝑥) ∈ V
96, 7, 8fvmpt 6241 . . . . . . . 8 (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥))
105, 9ax-mp 5 . . . . . . 7 ((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥)
11 fveqres 6189 . . . . . . 7 (((𝑧 ∈ V ↦ (𝐹𝑧))‘𝑥) = (𝐹𝑥) → (((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹𝐴)‘𝑥))
1210, 11ax-mp 5 . . . . . 6 (((𝑧 ∈ V ↦ (𝐹𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹𝐴)‘𝑥)
134, 12eqtr3i 2645 . . . . 5 ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) = ((𝐹𝐴)‘𝑥)
1413eqeq2i 2633 . . . 4 (𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) ↔ 𝑦 = ((𝐹𝐴)‘𝑥))
1514exbii 1771 . . 3 (∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥) ↔ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥))
1615abbii 2736 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} = {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)}
17 fvresex.1 . . . 4 𝐴 ∈ V
1817mptex 6443 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) ∈ V
1918fvclex 7088 . 2 {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧𝐴 ↦ (𝐹𝑧))‘𝑥)} ∈ V
2016, 19eqeltrri 2695 1 {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹𝐴)‘𝑥)} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wex 1701  wcel 1987  {cab 2607  Vcvv 3186  wss 3556  cmpt 4675  cres 5078  cfv 5849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857
This theorem is referenced by: (None)
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