Theorem brovmptimex 44753

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 5124	. 2 ⊢ (𝜑 → 𝐴(𝐶𝐹𝐷)𝐵)
4		brne0 5161	. 2 ⊢ (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅)
5		brovmptimex.mpt	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)
6	5	reldmmpo 7544	. . . 4 ⊢ Rel dom 𝐹
7	6	ovprc 7448	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅)
8	7	necon1ai 2985	. 2 ⊢ ((𝐶𝐹𝐷) ≠ ∅ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	3, 4, 8	3syl 19	1 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4286   class class class wbr 5109  (class class class)co 7410   ∈ cmpo 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-dm 5671  df-iota 6492  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415

This theorem is referenced by:  brovmptimex1  44754  brovmptimex2  44755