Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brovmptimex Structured version   Visualization version   GIF version

Theorem brovmptimex 37345
Description: If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)
Hypotheses
Ref Expression
brovmptimex.mpt 𝐹 = (𝑥𝐸, 𝑦𝐺𝐻)
brovmptimex.br (𝜑𝐴𝑅𝐵)
brovmptimex.ov (𝜑𝑅 = (𝐶𝐹𝐷))
Assertion
Ref Expression
brovmptimex (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))
Distinct variable groups:   𝑥,𝐸,𝑦   𝑦,𝐹
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem brovmptimex
StepHypRef Expression
1 brovmptimex.ov . . 3 (𝜑𝑅 = (𝐶𝐹𝐷))
2 brovmptimex.br . . 3 (𝜑𝐴𝑅𝐵)
31, 2breqdi 4598 . 2 (𝜑𝐴(𝐶𝐹𝐷)𝐵)
4 brne0 4632 . 2 (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅)
5 brovmptimex.mpt . . . . 5 𝐹 = (𝑥𝐸, 𝑦𝐺𝐻)
65reldmmpt2 6669 . . . 4 Rel dom 𝐹
76ovprc 6581 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅)
87necon1ai 2809 . 2 ((𝐶𝐹𝐷) ≠ ∅ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
93, 4, 83syl 18 1 (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  c0 3874   class class class wbr 4583  (class class class)co 6549  cmpt2 6551
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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
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-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554
This theorem is referenced by:  brovmptimex1  37346  brovmptimex2  37347
  Copyright terms: Public domain W3C validator