Theorem relbigcup 35878

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 5649	. . 3 ⊢ Rel (V × V)
2		reldif 5769	. . 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 35839	. . 3 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
5	4	releqi 5732	. 2 ⊢ (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
6	3, 5	mpbir 231	1 ⊢ Rel Bigcup

Syntax hints:  Vcvv 3444   ∖ cdif 3908   △ csymdif 4211   E cep 5530   × cxp 5629  ran crn 5632   ∘ ccom 5635  Rel wrel 5636   ⊗ ctxp 35811   Bigcup cbigcup 35815

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-dif 3914  df-ss 3928  df-opab 5165  df-xp 5637  df-rel 5638  df-bigcup 35839

This theorem is referenced by:  brbigcup  35879  dfbigcup2  35880