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 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)
brovmptimex.br	⊢ (𝜑 → 𝐴𝑅𝐵)
brovmptimex.ov	⊢ (𝜑 → 𝑅 = (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

Step	Hyp	Ref	Expression
1		brovmptimex.ov	. . 3 ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))
2		brovmptimex.br	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
3	1, 2	breqdi 4799	. 2 ⊢ (𝜑 → 𝐴(𝐶𝐹𝐷)𝐵)
4		brne0 4834	. 2 ⊢ (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅)
5		brovmptimex.mpt	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)
6	5	reldmmpt2 6917	. . . 4 ⊢ Rel dom 𝐹
7	6	ovprc 6827	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅)
8	7	necon1ai 2969	. 2 ⊢ ((𝐶𝐹𝐷) ≠ ∅ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	3, 4, 8	3syl 18	1 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))

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