Theorem brovmptimex 45012

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 5118	. 2 ⊢ (𝜑 → 𝐴(𝐶𝐹𝐷)𝐵)
4		brne0 5155	. 2 ⊢ (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅)
5		brovmptimex.mpt	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)
6	5	reldmmpo 7552	. . . 4 ⊢ Rel dom 𝐹
7	6	ovprc 7456	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅)
8	7	necon1ai 2983	. 2 ⊢ ((𝐶𝐹𝐷) ≠ ∅ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	3, 4, 8	3syl 19	1 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  Vcvv 3451  ∅c0 4279   class class class wbr 5103  (class class class)co 7418   ∈ cmpo 7420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dm 5661  df-iota 6493  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423

This theorem is used by:  brovmptimex1  45013  brovmptimex2  45014