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Theorem fabex 7615
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 7614 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
51, 2, 4mp2an 691 1 𝐹 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  {cab 2799  Vcvv 3471  wf 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-xp 5534  df-rel 5535  df-cnv 5536  df-dm 5538  df-rn 5539  df-fun 6330  df-fn 6331  df-f 6332
This theorem is referenced by: (None)
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