Theorem brovmptimex2 43269

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)

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 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 43267	. 2 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))
5	4	simprd 495	1 ⊢ (𝜑 → 𝐷 ∈ V)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3466   class class class wbr 5138  (class class class)co 7401   ∈ cmpo 7403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-dm 5676  df-iota 6485  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406

This theorem is referenced by:  ntrneibex  43313