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Theorem opabex 7171
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 6537 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
2 opabex.2 . . . 4 (𝑥𝐴 → ∃*𝑦𝜑)
3 moanimv 2616 . . . 4 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
42, 3mpbir 230 . . 3 ∃*𝑦(𝑥𝐴𝜑)
51, 4mpgbir 1802 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
6 opabex.1 . . 3 𝐴 ∈ V
7 dmopabss 5875 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
86, 7ssexi 5280 . 2 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
9 funex 7170 . 2 ((Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V)
105, 8, 9mp2an 691 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  ∃*wmo 2533  Vcvv 3444  {copab 5168  dom cdm 5634  Fun wfun 6491
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by: (None)
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