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Theorem fabexg 7925
Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
Hypothesis
Ref Expression
fabexg.1 𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}
Assertion
Ref Expression
fabexg ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝐹(𝑥)

Proof of Theorem fabexg
StepHypRef Expression
1 xpexg 7737 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
2 pwexg 5377 . 2 ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
3 fabexg.1 . . . . 5 𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}
4 fssxp 6746 . . . . . . . 8 (𝑥:𝐴𝐵𝑥 ⊆ (𝐴 × 𝐵))
5 velpw 4608 . . . . . . . 8 (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑥 ⊆ (𝐴 × 𝐵))
64, 5sylibr 233 . . . . . . 7 (𝑥:𝐴𝐵𝑥 ∈ 𝒫 (𝐴 × 𝐵))
76anim1i 616 . . . . . 6 ((𝑥:𝐴𝐵𝜑) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑))
87ss2abi 4064 . . . . 5 {𝑥 ∣ (𝑥:𝐴𝐵𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
93, 8eqsstri 4017 . . . 4 𝐹 ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
10 ssab2 4077 . . . 4 {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} ⊆ 𝒫 (𝐴 × 𝐵)
119, 10sstri 3992 . . 3 𝐹 ⊆ 𝒫 (𝐴 × 𝐵)
12 ssexg 5324 . . 3 ((𝐹 ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → 𝐹 ∈ V)
1311, 12mpan 689 . 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 397   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3475  wss 3949  𝒫 cpw 4603   × cxp 5675  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  fabex  7926  f1oabexg  7927  elghomlem1OLD  36753
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