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Theorem fabex 7629
Description: Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
Hypotheses
Ref Expression
fabex.1 𝐴 ∈ V
fabex.2 𝐵 ∈ V
fabex.3 𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}
Assertion
Ref Expression
fabex 𝐹 ∈ V
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem fabex
StepHypRef Expression
1 fabex.1 . 2 𝐴 ∈ V
2 fabex.2 . 2 𝐵 ∈ V
3 fabex.3 . . 3 𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}
43fabexg 7628 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
51, 2, 4mp2an 688 1 𝐹 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  {cab 2796  Vcvv 3492  wf 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-fun 6350  df-fn 6351  df-f 6352
This theorem is referenced by: (None)
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