Theorem fabex 7876

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 7874	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
5	1, 2, 4	mp2an 692	1 ⊢ 𝐹 ∈ V

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  Vcvv 3437  ⟶wf 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-fun 6488  df-fn 6489  df-f 6490

This theorem is referenced by: (None)