Theorem relbigcup 36286

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 5680	. . 3 ⊢ Rel (V × V)
2		reldif 5803	. . 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 36247	. . 3 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
5	4	releqi 5765	. 2 ⊢ (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))))
6	3, 5	mpbir 234	1 ⊢ Rel Bigcup

Syntax hints:  Vcvv 3463   ∖ cdif 3910   △ csymdif 4213   E cep 5561   × cxp 5660  ran crn 5663   ∘ ccom 5666  Rel wrel 5667   ⊗ ctxp 36219   Bigcup cbigcup 36223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-ss 3930  df-opab 5178  df-xp 5668  df-rel 5669  df-bigcup 36247

This theorem is referenced by:  brbigcup  36287  dfbigcup2  36288