Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opabresex0d Structured version   Visualization version   GIF version

Theorem opabresex0d 40330
 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 553 . . . 4 ((𝜑𝑥𝑅𝑦) → (𝑥𝐶𝜃))
43ex 449 . . 3 (𝜑 → (𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
54alrimivv 1843 . 2 (𝜑 → ∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)))
6 opabresex0d.c . . . 4 (𝜑𝐶𝑊)
76elexd 3187 . . 3 (𝜑𝐶 ∈ V)
8 opabresex0d.y . . . 4 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)
98elexd 3187 . . 3 ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ V)
107, 9opabex3d 7037 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V)
11 opabbrex 6593 . 2 ((∀𝑥𝑦(𝑥𝑅𝑦 → (𝑥𝐶𝜃)) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜃)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
125, 10, 11syl2anc 691 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  {cab 2596  Vcvv 3173   class class class wbr 4583  {copab 4642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  opabbrfex0d  40331  opabresexd  40332
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