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Theorem brovmptimex1 40718
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
brovmptimex1 (𝜑𝐶 ∈ 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 brovmptimex1
StepHypRef Expression
1 brovmptimex.mpt . . 3 𝐹 = (𝑥𝐸, 𝑦𝐺𝐻)
2 brovmptimex.br . . 3 (𝜑𝐴𝑅𝐵)
3 brovmptimex.ov . . 3 (𝜑𝑅 = (𝐶𝐹𝐷))
41, 2, 3brovmptimex 40717 . 2 (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))
54simpld 498 1 (𝜑𝐶 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  Vcvv 3444   class class class wbr 5033  (class class class)co 7139  cmpo 7141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-dm 5533  df-iota 6287  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144
This theorem is referenced by: (None)
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