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Theorem fabexg 7628
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 7462 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
2 pwexg 5270 . 2 ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
3 fabexg.1 . . . . 5 𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}
4 fssxp 6527 . . . . . . . 8 (𝑥:𝐴𝐵𝑥 ⊆ (𝐴 × 𝐵))
5 velpw 4543 . . . . . . . 8 (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑥 ⊆ (𝐴 × 𝐵))
64, 5sylibr 235 . . . . . . 7 (𝑥:𝐴𝐵𝑥 ∈ 𝒫 (𝐴 × 𝐵))
76anim1i 614 . . . . . 6 ((𝑥:𝐴𝐵𝜑) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑))
87ss2abi 4040 . . . . 5 {𝑥 ∣ (𝑥:𝐴𝐵𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
93, 8eqsstri 3998 . . . 4 𝐹 ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
10 ssab2 4052 . . . 4 {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} ⊆ 𝒫 (𝐴 × 𝐵)
119, 10sstri 3973 . . 3 𝐹 ⊆ 𝒫 (𝐴 × 𝐵)
12 ssexg 5218 . . 3 ((𝐹 ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → 𝐹 ∈ V)
1311, 12mpan 686 . 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 396   = wceq 1528  wcel 2105  {cab 2796  Vcvv 3492  wss 3933  𝒫 cpw 4535   × cxp 5546  wf 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-fun 6350  df-fn 6351  df-f 6352
This theorem is referenced by:  fabex  7629  f1oabexg  7631  elghomlem1OLD  35044
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