Theorem relbigcup 34248

Description: The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)

Assertion

Ref	Expression
relbigcup	⊢ Rel Bigcup

Proof of Theorem relbigcup

Step	Hyp	Ref	Expression
1		relxp 5618	. . 3 ⊢ Rel (V × V)
2		reldif 5737	. . 3 ⊢ (Rel (V × V) → Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
3	1, 2	ax-mp 5	. 2 ⊢ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
4		df-bigcup 34209	. . 3 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
5	4	releqi 5699	. 2 ⊢ (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
6	3, 5	mpbir 230	1 ⊢ Rel Bigcup

Syntax hints:  Vcvv 3437   ∖ cdif 3889   △ csymdif 4181   E cep 5505   × cxp 5598  ran crn 5601   ∘ ccom 5604  Rel wrel 5605   ⊗ ctxp 34181   Bigcup cbigcup 34185

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-dif 3895  df-in 3899  df-ss 3909  df-opab 5144  df-xp 5606  df-rel 5607  df-bigcup 34209

This theorem is referenced by:  brbigcup  34249  dfbigcup2  34250