Theorem fixun 32016

Description: The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
fixun	|- Fix ( A u. B ) = ( Fix A u. Fix B )

Proof of Theorem fixun

Step	Hyp	Ref	Expression
1		indir 3875	. . . 4 |- ( ( A u. B ) i^i _I ) = ( ( A i^i _I ) u. ( B i^i _I ) )
2	1	dmeqi 5325	. . 3 |- dom ( ( A u. B ) i^i _I ) = dom ( ( A i^i _I ) u. ( B i^i _I ) )
3		dmun 5331	. . 3 |- dom ( ( A i^i _I ) u. ( B i^i _I ) ) = ( dom ( A i^i _I ) u. dom ( B i^i _I ) )
4	2, 3	eqtri 2644	. 2 |- dom ( ( A u. B ) i^i _I ) = ( dom ( A i^i _I ) u. dom ( B i^i _I ) )
5		df-fix 31966	. 2 |- Fix ( A u. B ) = dom ( ( A u. B ) i^i _I )
6		df-fix 31966	. . 3 |- Fix A = dom ( A i^i _I )
7		df-fix 31966	. . 3 |- Fix B = dom ( B i^i _I )
8	6, 7	uneq12i 3765	. 2 |- ( Fix A u. Fix B ) = ( dom ( A i^i _I ) u. dom ( B i^i _I ) )
9	4, 5, 8	3eqtr4i 2654	1 |- Fix ( A u. B ) = ( Fix A u. Fix B )

Syntax hints:   = wceq 1483   u. cun 3572   i^i cin 3573   _I cid 5023   dom cdm 5114   Fixcfix 31942

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-3an 1039  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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-dm 5124  df-fix 31966

This theorem is referenced by: (None)