Theorem fabexg 7942

Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypothesis

Ref	Expression
fabexg.1	⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}

Assertion

Ref	Expression
fabexg	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem fabexg

Step	Hyp	Ref	Expression
1		xpexg 7752	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
2		pwexg 5378	. 2 ⊢ ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
3		fabexg.1	. . . . 5 ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}
4		fssxp 6751	. . . . . . . 8 ⊢ (𝑥:𝐴⟶𝐵 → 𝑥 ⊆ (𝐴 × 𝐵))
5		velpw 4608	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑥 ⊆ (𝐴 × 𝐵))
6	4, 5	sylibr 233	. . . . . . 7 ⊢ (𝑥:𝐴⟶𝐵 → 𝑥 ∈ 𝒫 (𝐴 × 𝐵))
7	6	anim1i 614	. . . . . 6 ⊢ ((𝑥:𝐴⟶𝐵 ∧ 𝜑) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑))
8	7	ss2abi 4061	. . . . 5 ⊢ {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
9	3, 8	eqsstri 4014	. . . 4 ⊢ 𝐹 ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
10		ssab2 4074	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} ⊆ 𝒫 (𝐴 × 𝐵)
11	9, 10	sstri 3989	. . 3 ⊢ 𝐹 ⊆ 𝒫 (𝐴 × 𝐵)
12		ssexg 5323	. . 3 ⊢ ((𝐹 ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → 𝐹 ∈ V)
13	11, 12	mpan 689	. 2 ⊢ (𝒫 (𝐴 × 𝐵) ∈ V → 𝐹 ∈ V)
14	1, 2, 13	3syl 18	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705  Vcvv 3471   ⊆ wss 3947  𝒫 cpw 4603   × cxp 5676  ⟶wf 6544

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 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

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 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-fun 6550  df-fn 6551  df-f 6552

This theorem is referenced by:  fabex  7943  f1oabexg  7944  elghomlem1OLD  37358