Theorem fabex 7962

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:𝐴⟶𝐵 ∧ 𝜑)}

Assertion

Ref	Expression
fabex	⊢ 𝐹 ∈ V

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem fabex

Step	Hyp	Ref	Expression
1		fabex.1	. 2 ⊢ 𝐴 ∈ V
2		fabex.2	. 2 ⊢ 𝐵 ∈ V
3		fabex.3	. . 3 ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}
4	3	fabexg 7960	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
5	1, 2, 4	mp2an 692	1 ⊢ 𝐹 ∈ V

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480  ⟶wf 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565

This theorem is referenced by: (None)