Theorem relbigcup 34858

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 5694	. . 3 ⊢ Rel (V × V)
2		reldif 5814	. . 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 34819	. . 3 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
5	4	releqi 5776	. 2 ⊢ (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
6	3, 5	mpbir 230	1 ⊢ Rel Bigcup

Syntax hints:  Vcvv 3475   ∖ cdif 3945   △ csymdif 4241   E cep 5579   × cxp 5674  ran crn 5677   ∘ ccom 5680  Rel wrel 5681   ⊗ ctxp 34791   Bigcup cbigcup 34795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3951  df-in 3955  df-ss 3965  df-opab 5211  df-xp 5682  df-rel 5683  df-bigcup 34819

This theorem is referenced by:  brbigcup  34859  dfbigcup2  34860