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Theorem opabresex0d 45993
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 513 . . . 4 ((𝜑𝑥𝑅𝑦) → (𝑥𝐶𝜃))
43ex 414 . . 3 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
54alrimivv 1932 . 2 (𝜑 → ∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
6 opabresex0d.c . . 3 (𝜑𝐶𝑊)
7 opabresex0d.y . . . 4 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)
87elexd 3495 . . 3 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ V)
96, 8opabex3d 7952 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V)
10 opabbrex 7460 . 2 ((∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
115, 9, 10syl2anc 585 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wcel 2107  {cab 2710  Vcvv 3475   class class class wbr 5149  {copab 5211
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-opab 5212  df-xp 5683  df-rel 5684
This theorem is referenced by:  opabbrfex0d  45994  opabresexd  45995
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