Theorem bj-projun 32982

Description: The class projection on a given component preserves unions. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-projun	|- ( A Proj ( B u. C ) ) = ( ( A Proj B ) u. ( A Proj C ) )

Proof of Theorem bj-projun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-proj 32979	. . . . 5 |- ( A Proj B ) = { x | { x } e. ( B " { A } ) }
2	1	abeq2i 2735	. . . 4 |- ( x e. ( A Proj B ) <-> { x } e. ( B " { A } ) )
3		df-bj-proj 32979	. . . . 5 |- ( A Proj C ) = { x | { x } e. ( C " { A } ) }
4	3	abeq2i 2735	. . . 4 |- ( x e. ( A Proj C ) <-> { x } e. ( C " { A } ) )
5	2, 4	orbi12i 543	. . 3 |- ( ( x e. ( A Proj B ) \/ x e. ( A Proj C ) ) <-> ( { x } e. ( B " { A } ) \/ { x } e. ( C " { A } ) ) )
6		elun 3753	. . 3 |- ( x e. ( ( A Proj B ) u. ( A Proj C ) ) <-> ( x e. ( A Proj B ) \/ x e. ( A Proj C ) ) )
7		df-bj-proj 32979	. . . . 5 |- ( A Proj ( B u. C ) ) = { x | { x } e. ( ( B u. C ) " { A } ) }
8	7	abeq2i 2735	. . . 4 |- ( x e. ( A Proj ( B u. C ) ) <-> { x } e. ( ( B u. C ) " { A } ) )
9		imaundir 5546	. . . . 5 |- ( ( B u. C ) " { A } ) = ( ( B " { A } ) u. ( C " { A } ) )
10	9	eleq2i 2693	. . . 4 |- ( { x } e. ( ( B u. C ) " { A } ) <-> { x } e. ( ( B " { A } ) u. ( C " { A } ) ) )
11		elun 3753	. . . 4 |- ( { x } e. ( ( B " { A } ) u. ( C " { A } ) ) <-> ( { x } e. ( B " { A } ) \/ { x } e. ( C " { A } ) ) )
12	8, 10, 11	3bitri 286	. . 3 |- ( x e. ( A Proj ( B u. C ) ) <-> ( { x } e. ( B " { A } ) \/ { x } e. ( C " { A } ) ) )
13	5, 6, 12	3bitr4ri 293	. 2 |- ( x e. ( A Proj ( B u. C ) ) <-> x e. ( ( A Proj B ) u. ( A Proj C ) ) )
14	13	eqriv 2619	1 |- ( A Proj ( B u. C ) ) = ( ( A Proj B ) u. ( A Proj C ) )

Syntax hints:   \/ wo 383   = wceq 1483   e. wcel 1990   u. cun 3572   {csn 4177   "cima 5117   Proj bj-cproj 32978

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-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-bj-proj 32979

This theorem is referenced by:  bj-pr1un  32991  bj-pr2un  33005