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Theorem fabex 7937
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 7936 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
51, 2, 4mp2an 691 1 𝐹 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  {cab 2705  Vcvv 3470  wf 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-fun 6544  df-fn 6545  df-f 6546
This theorem is referenced by: (None)
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