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Theorem relun 5473
 Description: The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
relun (Rel (𝐴𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))

Proof of Theorem relun
StepHypRef Expression
1 unss 4016 . 2 ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴𝐵) ⊆ (V × V))
2 df-rel 5353 . . 3 (Rel 𝐴𝐴 ⊆ (V × V))
3 df-rel 5353 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
42, 3anbi12i 620 . 2 ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)))
5 df-rel 5353 . 2 (Rel (𝐴𝐵) ↔ (𝐴𝐵) ⊆ (V × V))
61, 4, 53bitr4ri 296 1 (Rel (𝐴𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386  Vcvv 3414   ∪ cun 3796   ⊆ wss 3798   × cxp 5344  Rel wrel 5351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-un 3803  df-in 3805  df-ss 3812  df-rel 5353 This theorem is referenced by:  difxp  5803  funun  6172  fununfun  6174
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