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Theorem opabex 5761
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 5270 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
2 opabex.2 . . . 4 (𝑥𝐴 → ∃*𝑦𝜑)
3 moanimv 2113 . . . 4 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
42, 3mpbir 146 . . 3 ∃*𝑦(𝑥𝐴𝜑)
51, 4mpgbir 1464 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
6 opabex.1 . . 3 𝐴 ∈ V
7 dmopabss 4857 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
86, 7ssexi 4156 . 2 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
9 funex 5760 . 2 ((Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V)
105, 8, 9mp2an 426 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  ∃*wmo 2039  wcel 2160  Vcvv 2752  {copab 4078  dom cdm 4644  Fun wfun 5229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243
This theorem is referenced by: (None)
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