Theorem brun 5170

Description: The union of two binary relations. (Contributed by NM, 21-Dec-2008.)

Assertion

Ref	Expression
brun	⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵))

Proof of Theorem brun

Step	Hyp	Ref	Expression
1		elun 4128	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∪ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∨ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 5120	. 2 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∪ 𝑆))
3		df-br 5120	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5120	. . 3 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	3, 4	orbi12i 914	. 2 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∨ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
6	1, 2, 5	3bitr4i 303	1 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵))

Syntax hints:   ↔ wb 206   ∨ wo 847   ∈ wcel 2108   ∪ cun 3924  ⟨cop 4607   class class class wbr 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-br 5120

This theorem is referenced by:  dmun  5890  qfto  6110  poleloe  6120  cnvun  6131  coundi  6236  coundir  6237  fununmo  6582  eqfunresadj  7352  brdifun  8747  fpwwe2lem12  10654  ltxrlt  11303  ltxr  13129  dfle2  13161  brprop  32620  satfbrsuc  35334  dfso2  35718  dfon3  35856  brcup  35903  dfrdg4  35915  dffrege99  43933