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Theorem opabbrfex0d 46579
Description: A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.)
Hypotheses
Ref Expression
opabresex0d.x ((𝜑𝑥𝑅𝑦) → 𝑥𝐶)
opabresex0d.t ((𝜑𝑥𝑅𝑦) → 𝜃)
opabresex0d.y ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)
opabresex0d.c (𝜑𝐶𝑊)
Assertion
Ref Expression
opabbrfex0d (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)
Distinct variable groups:   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜃(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabbrfex0d
StepHypRef Expression
1 pm4.24 563 . . 3 (𝑥𝑅𝑦 ↔ (𝑥𝑅𝑦𝑥𝑅𝑦))
21opabbii 5209 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑥𝑅𝑦)}
3 opabresex0d.x . . 3 ((𝜑𝑥𝑅𝑦) → 𝑥𝐶)
4 opabresex0d.t . . 3 ((𝜑𝑥𝑅𝑦) → 𝜃)
5 opabresex0d.y . . 3 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)
6 opabresex0d.c . . 3 (𝜑𝐶𝑊)
73, 4, 5, 6opabresex0d 46578 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑥𝑅𝑦)} ∈ V)
82, 7eqeltrid 2832 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  {cab 2704  Vcvv 3469   class class class wbr 5142  {copab 5204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-opab 5205  df-xp 5678  df-rel 5679
This theorem is referenced by: (None)
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