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Theorem fabexg 7960
Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.) (Proof shortened by AV, 9-Jun-2025.)
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 elex 3501 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3501 . 2 (𝐵𝐷𝐵 ∈ V)
3 fabexg.1 . . 3 𝐹 = {𝑥 ∣ (𝑥:𝐴𝐵𝜑)}
4 simprl 771 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝑥:𝐴𝐵𝜑)) → 𝑥:𝐴𝐵)
5 simpl 482 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
6 simpr 484 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
74, 5, 6fabexd 7959 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ (𝑥:𝐴𝐵𝜑)} ∈ V)
83, 7eqeltrid 2845 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
91, 2, 8syl2an 596 1 ((𝐴𝐶𝐵𝐷) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  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:  fabex  7962  mapex  7963  f1oabexgOLD  7965  elghomlem1OLD  37892
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