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Theorem brovmptimex 38844
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 4799 . 2 (𝜑𝐴(𝐶𝐹𝐷)𝐵)
4 brne0 4834 . 2 (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅)
5 brovmptimex.mpt . . . . 5 𝐹 = (𝑥𝐸, 𝑦𝐺𝐻)
65reldmmpt2 6917 . . . 4 Rel dom 𝐹
76ovprc 6827 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅)
87necon1ai 2969 . 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 382   = wceq 1630  wcel 2144  wne 2942  Vcvv 3349  c0 4061   class class class wbr 4784  (class class class)co 6792  cmpt2 6794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-dm 5259  df-iota 5994  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797
This theorem is referenced by:  brovmptimex1  38845  brovmptimex2  38846
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