Theorem fabex 7887

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 7885	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
5	1, 2, 4	mp2an 698	1 ⊢ 𝐹 ∈ V

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2718  Vcvv 3432  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by: (None)