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Theorem fabex 7873
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 7872 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
51, 2, 4mp2an 691 1 𝐹 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3444  wf 6493
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-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by: (None)
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