Theorem fabexg 7015

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 6858	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
2		pwexg 4776	. 2 ⊢ ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
3		fabexg.1	. . . . 5 ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}
4		fssxp 5973	. . . . . . . 8 ⊢ (𝑥:𝐴⟶𝐵 → 𝑥 ⊆ (𝐴 × 𝐵))
5		selpw 4115	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑥 ⊆ (𝐴 × 𝐵))
6	4, 5	sylibr 223	. . . . . . 7 ⊢ (𝑥:𝐴⟶𝐵 → 𝑥 ∈ 𝒫 (𝐴 × 𝐵))
7	6	anim1i 590	. . . . . 6 ⊢ ((𝑥:𝐴⟶𝐵 ∧ 𝜑) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑))
8	7	ss2abi 3637	. . . . 5 ⊢ {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
9	3, 8	eqsstri 3598	. . . 4 ⊢ 𝐹 ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
10		ssab2 3649	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} ⊆ 𝒫 (𝐴 × 𝐵)
11	9, 10	sstri 3577	. . 3 ⊢ 𝐹 ⊆ 𝒫 (𝐴 × 𝐵)
12		ssexg 4732	. . 3 ⊢ ((𝐹 ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → 𝐹 ∈ V)
13	11, 12	mpan 702	. 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 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   × cxp 5036  ⟶wf 5800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808

This theorem is referenced by:  fabex  7016  f1oabexg  7018  elghomlem1OLD  32854