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Theorem brovmptimex 44017
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 5163 . 2 (𝜑𝐴(𝐶𝐹𝐷)𝐵)
4 brne0 5198 . 2 (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅)
5 brovmptimex.mpt . . . . 5 𝐹 = (𝑥𝐸, 𝑦𝐺𝐻)
65reldmmpo 7567 . . . 4 Rel dom 𝐹
76ovprc 7469 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅)
87necon1ai 2966 . 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 395   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  c0 4339   class class class wbr 5148  (class class class)co 7431  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  brovmptimex1  44018  brovmptimex2  44019
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