Theorem fabexgOLD 7961

Description: Obsolete version of fabexg 7960 as of 9-Jun-2025. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
fabexgOLD	⊢ ((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 fabexgOLD

Step	Hyp	Ref	Expression
1		xpexg 7770	. 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 6763	. . . . . . . 8 ⊢ (𝑥:𝐴⟶𝐵 → 𝑥 ⊆ (𝐴 × 𝐵))
5		velpw 4605	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑥 ⊆ (𝐴 × 𝐵))
6	4, 5	sylibr 234	. . . . . . 7 ⊢ (𝑥:𝐴⟶𝐵 → 𝑥 ∈ 𝒫 (𝐴 × 𝐵))
7	6	anim1i 615	. . . . . 6 ⊢ ((𝑥:𝐴⟶𝐵 ∧ 𝜑) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑))
8	7	ss2abi 4067	. . . . 5 ⊢ {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
9	3, 8	eqsstri 4030	. . . 4 ⊢ 𝐹 ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
10		ssab2 4079	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} ⊆ 𝒫 (𝐴 × 𝐵)
11	9, 10	sstri 3993	. . 3 ⊢ 𝐹 ⊆ 𝒫 (𝐴 × 𝐵)
12		ssexg 5323	. . 3 ⊢ ((𝐹 ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → 𝐹 ∈ V)
13	11, 12	mpan 690	. 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 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480   ⊆ wss 3951  𝒫 cpw 4600   × cxp 5683  ⟶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)