Theorem brovmptimex2 45290

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
brovmptimex2	⊢ (𝜑 → 𝐷 ∈ V)

Proof of Theorem brovmptimex2

Step	Hyp	Ref	Expression
1		brovmptimex.mpt	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)
2		brovmptimex.br	. . 3 ⊢ (𝜑 → 𝐴𝑅𝐵)
3		brovmptimex.ov	. . 3 ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))
4	1, 2, 3	brovmptimex 45288	. 2 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))
5	4	simprd 501	1 ⊢ (𝜑 → 𝐷 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   class class class wbr 5108  (class class class)co 7451   ∈ cmpo 7453

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-dm 5668  df-iota 6520  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456

This theorem is used by:  ntrneibex  45334