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Theorem opabresex0d 43361
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 512 . . . 4 ((𝜑𝑥𝑅𝑦) → (𝑥𝐶𝜃))
43ex 413 . . 3 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
54alrimivv 1920 . 2 (𝜑 → ∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
6 opabresex0d.c . . . 4 (𝜑𝐶𝑊)
76elexd 3512 . . 3 (𝜑𝐶 ∈ V)
8 opabresex0d.y . . . 4 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)
98elexd 3512 . . 3 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ V)
107, 9opabex3d 7655 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V)
11 opabbrex 7196 . 2 ((∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
125, 10, 11syl2anc 584 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526  wcel 2105  {cab 2796  Vcvv 3492   class class class wbr 5057  {copab 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  opabbrfex0d  43362  opabresexd  43363
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