MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fabexgOLD Structured version   Visualization version   GIF version

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:𝐴𝐵𝜑)}
Assertion
Ref Expression
fabexgOLD ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝐹(𝑥)

Proof of Theorem fabexgOLD
StepHypRef Expression
1 xpexg 7770 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
2 pwexg 5378 . 2 ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
3 fabexg.1 . . . . 5 𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}
4 fssxp 6763 . . . . . . . 8 (𝑥:𝐴𝐵𝑥 ⊆ (𝐴 × 𝐵))
5 velpw 4605 . . . . . . . 8 (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑥 ⊆ (𝐴 × 𝐵))
64, 5sylibr 234 . . . . . . 7 (𝑥:𝐴𝐵𝑥 ∈ 𝒫 (𝐴 × 𝐵))
76anim1i 615 . . . . . 6 ((𝑥:𝐴𝐵𝜑) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑))
87ss2abi 4067 . . . . 5 {𝑥 ∣ (𝑥:𝐴𝐵𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
93, 8eqsstri 4030 . . . 4 𝐹 ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
10 ssab2 4079 . . . 4 {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} ⊆ 𝒫 (𝐴 × 𝐵)
119, 10sstri 3993 . . 3 𝐹 ⊆ 𝒫 (𝐴 × 𝐵)
12 ssexg 5323 . . 3 ((𝐹 ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → 𝐹 ∈ V)
1311, 12mpan 690 . 2 (𝒫 (𝐴 × 𝐵) ∈ V → 𝐹 ∈ V)
141, 2, 133syl 18 1 ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Colors of variables: wff setvar class
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)
  Copyright terms: Public domain W3C validator