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Theorem relbigcup 33242
 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 5571 . . 3 Rel (V × V)
2 reldif 5686 . . 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 33203 . . 3 Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
54releqi 5650 . 2 (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
63, 5mpbir 232 1 Rel Bigcup
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3499   ∖ cdif 3936   △ csymdif 4221   E cep 5462   × cxp 5551  ran crn 5554   ∘ ccom 5557  Rel wrel 5558   ⊗ ctxp 33175   Bigcup cbigcup 33179 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-v 3501  df-dif 3942  df-in 3946  df-ss 3955  df-opab 5125  df-xp 5559  df-rel 5560  df-bigcup 33203 This theorem is referenced by:  brbigcup  33243  dfbigcup2  33244
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