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Theorem opabex 6365
Description: Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
Hypotheses
Ref Expression
opabex.1 𝐴 ∈ V
opabex.2 (𝑥𝐴 → ∃*𝑦𝜑)
Assertion
Ref Expression
opabex {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabex
StepHypRef Expression
1 funopab 5822 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
2 opabex.2 . . . 4 (𝑥𝐴 → ∃*𝑦𝜑)
3 moanimv 2518 . . . 4 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
42, 3mpbir 219 . . 3 ∃*𝑦(𝑥𝐴𝜑)
51, 4mpgbir 1716 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
6 opabex.1 . . 3 𝐴 ∈ V
7 dmopabss 5244 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
86, 7ssexi 4725 . 2 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
9 funex 6364 . 2 ((Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V)
105, 8, 9mp2an 703 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  ∃*wmo 2458  Vcvv 3172  {copab 4636  dom cdm 5027  Fun wfun 5783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797
This theorem is referenced by: (None)
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