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Theorem fabexg 7500
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 7335 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 × 𝐵) ∈ V)
2 pwexg 5175 . 2 ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
3 fabexg.1 . . . . 5 𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}
4 fssxp 6407 . . . . . . . 8 (𝑥:𝐴𝐵𝑥 ⊆ (𝐴 × 𝐵))
5 selpw 4464 . . . . . . . 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 3968 . . . . 5 {𝑥 ∣ (𝑥:𝐴𝐵𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
93, 8eqsstri 3926 . . . 4 𝐹 ⊆ {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)}
10 ssab2 3980 . . . 4 {𝑥 ∣ (𝑥 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝜑)} ⊆ 𝒫 (𝐴 × 𝐵)
119, 10sstri 3902 . . 3 𝐹 ⊆ 𝒫 (𝐴 × 𝐵)
12 ssexg 5123 . . 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 1522  wcel 2081  {cab 2775  Vcvv 3437  wss 3863  𝒫 cpw 4457   × cxp 5446  wf 6226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-xp 5454  df-rel 5455  df-cnv 5456  df-dm 5458  df-rn 5459  df-fun 6232  df-fn 6233  df-f 6234
This theorem is referenced by:  fabex  7501  f1oabexg  7503  elghomlem1OLD  34720
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