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 41637
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 5089 . 2 (𝜑𝐴(𝐶𝐹𝐷)𝐵)
4 brne0 5124 . 2 (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅)
5 brovmptimex.mpt . . . . 5 𝐹 = (𝑥𝐸, 𝑦𝐺𝐻)
65reldmmpo 7408 . . . 4 Rel dom 𝐹
76ovprc 7313 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅)
87necon1ai 2971 . 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 396   = wceq 1539  wcel 2106  wne 2943  Vcvv 3432  c0 4256   class class class wbr 5074  (class class class)co 7275  cmpo 7277
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599  df-iota 6391  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280
This theorem is referenced by:  brovmptimex1  41638  brovmptimex2  41639
  Copyright terms: Public domain W3C validator