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Theorem relbigcup 31174
 Description: The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Assertion
Ref Expression
relbigcup Rel Bigcup

Proof of Theorem relbigcup
StepHypRef Expression
1 relxp 5150 . . 3 Rel (V × V)
2 reldif 5161 . . 3 (Rel (V × V) → Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
31, 2ax-mp 5 . 2 Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
4 df-bigcup 31134 . . 3 Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
54releqi 5125 . 2 (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
63, 5mpbir 220 1 Rel Bigcup
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3173   ∖ cdif 3537   △ csymdif 3805   E cep 4947   × cxp 5036  ran crn 5039   ∘ ccom 5042  Rel wrel 5043   ⊗ ctxp 31106   Bigcup cbigcup 31110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-opab 4644  df-xp 5044  df-rel 5045  df-bigcup 31134 This theorem is referenced by:  brbigcup  31175  dfbigcup2  31176
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