Theorem relbigcup 36038

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 5640	. . 3 ⊢ Rel (V × V)
2		reldif 5762	. . 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 35999	. . 3 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
5	4	releqi 5725	. 2 ⊢ (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
6	3, 5	mpbir 231	1 ⊢ Rel Bigcup

Syntax hints:  Vcvv 3438   ∖ cdif 3896   △ csymdif 4202   E cep 5521   × cxp 5620  ran crn 5623   ∘ ccom 5626  Rel wrel 5627   ⊗ ctxp 35971   Bigcup cbigcup 35975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-ss 3916  df-opab 5159  df-xp 5628  df-rel 5629  df-bigcup 35999

This theorem is referenced by:  brbigcup  36039  dfbigcup2  36040