Theorem relbigcup 32004

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 5227	. . 3 |- Rel ( _V X. _V )
2		reldif 5238	. . 3 |- ( Rel ( _V X. _V ) -> Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. _E ) (x) _V ) ) ) )
3	1, 2	ax-mp 5	. 2 |- Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. _E ) (x) _V ) ) )
4		df-bigcup 31965	. . 3 |- Bigcup = ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. _E ) (x) _V ) ) )
5	4	releqi 5202	. 2 |- ( Rel Bigcup <-> Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. _E ) (x) _V ) ) ) )
6	3, 5	mpbir 221	1 |- Rel Bigcup

Syntax hints:   _Vcvv 3200   \ cdif 3571   /_\ csymdif 3843   _E cep 5028   X. cxp 5112   ran crn 5115   o. ccom 5118   Rel wrel 5119   (x) ctxp 31937   Bigcupcbigcup 31941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-opab 4713  df-xp 5120  df-rel 5121  df-bigcup 31965

This theorem is referenced by:  brbigcup  32005  dfbigcup2  32006