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Theorem opabresex0d 46478
Description: A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.)
Hypotheses
Ref Expression
opabresex0d.x ((𝜑𝑥𝑅𝑦) → 𝑥𝐶)
opabresex0d.t ((𝜑𝑥𝑅𝑦) → 𝜃)
opabresex0d.y ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)
opabresex0d.c (𝜑𝐶𝑊)
Assertion
Ref Expression
opabresex0d (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Distinct variable groups:   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜃(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabresex0d
StepHypRef Expression
1 opabresex0d.x . . . . 5 ((𝜑𝑥𝑅𝑦) → 𝑥𝐶)
2 opabresex0d.t . . . . 5 ((𝜑𝑥𝑅𝑦) → 𝜃)
31, 2jca 511 . . . 4 ((𝜑𝑥𝑅𝑦) → (𝑥𝐶𝜃))
43ex 412 . . 3 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
54alrimivv 1923 . 2 (𝜑 → ∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
6 opabresex0d.c . . 3 (𝜑𝐶𝑊)
7 opabresex0d.y . . . 4 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)
87elexd 3487 . . 3 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ V)
96, 8opabex3d 7945 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V)
10 opabbrex 7452 . 2 ((∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
115, 9, 10syl2anc 583 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531  wcel 2098  {cab 2701  Vcvv 3466   class class class wbr 5138  {copab 5200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-opab 5201  df-xp 5672  df-rel 5673
This theorem is referenced by:  opabbrfex0d  46479  opabresexd  46480
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