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Theorem relbigcup 31681
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 5193 . . 3 Rel (V × V)
2 reldif 5204 . . 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 31641 . . 3 Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
54releqi 5168 . 2 (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
63, 5mpbir 221 1 Rel Bigcup
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3189  cdif 3556  csymdif 3826   E cep 4988   × cxp 5077  ran crn 5080  ccom 5083  Rel wrel 5084  ctxp 31613   Bigcup cbigcup 31617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-dif 3562  df-in 3566  df-ss 3573  df-opab 4679  df-xp 5085  df-rel 5086  df-bigcup 31641
This theorem is referenced by:  brbigcup  31682  dfbigcup2  31683
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