Theorem bj-projval 31635

Description: Value of the class projection (proof can be shortened by 19 bytes by using sylancl3 31209). (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-projval	|- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( B = A , C , (/) ) )

Proof of Theorem bj-projval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsncg 4003	. . . . . . . . 9 |- ( A e. V -> ( A e. { B } <-> A = B ) )
2		bj-xpima2sn 31596	. . . . . . . . 9 |- ( A e. { B } -> ( ( { B } X. tag C ) " { A } ) = tag C )
3	1, 2	syl6bir 237	. . . . . . . 8 |- ( A e. V -> ( A = B -> ( ( { B } X. tag C ) " { A } ) = tag C ) )
4	3	imp 435	. . . . . . 7 |- ( ( A e. V /\ A = B ) -> ( ( { B } X. tag C ) " { A } ) = tag C )
5	4	eleq2d 2525	. . . . . 6 |- ( ( A e. V /\ A = B ) -> ( { x } e. ( ( { B } X. tag C ) " { A } ) <-> { x } e. tag C ) )
6	5	abbidv 2580	. . . . 5 |- ( ( A e. V /\ A = B ) -> { x | { x } e. ( ( { B } X. tag C ) " { A } ) } = { x | { x } e. tag C } )
7		df-bj-proj 31630	. . . . 5 |- ( A Proj ( { B } X. tag C ) ) = { x | { x } e. ( ( { B } X. tag C ) " { A } ) }
8		bj-taginv 31625	. . . . 5 |- C = { x | { x } e. tag C }
9	6, 7, 8	3eqtr4g 2521	. . . 4 |- ( ( A e. V /\ A = B ) -> ( A Proj ( { B } X. tag C ) ) = C )
10	9	ex 440	. . 3 |- ( A e. V -> ( A = B -> ( A Proj ( { B } X. tag C ) ) = C ) )
11		noel 3747	. . . . 5 |- -. { x } e. (/)
12	7	abeq2i 2574	. . . . . 6 |- ( x e. ( A Proj ( { B } X. tag C ) ) <-> { x } e. ( ( { B } X. tag C ) " { A } ) )
13		elsni 4005	. . . . . . . . . 10 |- ( A e. { B } -> A = B )
14	13	con3i 142	. . . . . . . . 9 |- ( -. A = B -> -. A e. { B } )
15		df-nel 2636	. . . . . . . . 9 |- ( A e/ { B } <-> -. A e. { B } )
16	14, 15	sylibr 217	. . . . . . . 8 |- ( -. A = B -> A e/ { B } )
17		bj-xpima1sn 31594	. . . . . . . 8 |- ( A e/ { B } -> ( ( { B } X. tag C ) " { A } ) = (/) )
18	16, 17	syl 17	. . . . . . 7 |- ( -. A = B -> ( ( { B } X. tag C ) " { A } ) = (/) )
19	18	eleq2d 2525	. . . . . 6 |- ( -. A = B -> ( { x } e. ( ( { B } X. tag C ) " { A } ) <-> { x } e. (/) ) )
20	12, 19	syl5bb 265	. . . . 5 |- ( -. A = B -> ( x e. ( A Proj ( { B } X. tag C ) ) <-> { x } e. (/) ) )
21	11, 20	mtbiri 309	. . . 4 |- ( -. A = B -> -. x e. ( A Proj ( { B } X. tag C ) ) )
22	21	eq0rdv 3781	. . 3 |- ( -. A = B -> ( A Proj ( { B } X. tag C ) ) = (/) )
23		ifval 3932	. . 3 |- ( ( A Proj ( { B } X. tag C ) ) = if ( A = B , C , (/) ) <-> ( ( A = B -> ( A Proj ( { B } X. tag C ) ) = C ) /\ ( -. A = B -> ( A Proj ( { B } X. tag C ) ) = (/) ) ) )
24	10, 22, 23	sylancl3 31209	. 2 |- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( A = B , C , (/) ) )
25		eqcom 2469	. . 3 |- ( A = B <-> B = A )
26		ifbi 3914	. . 3 |- ( ( A = B <-> B = A ) -> if ( A = B , C , (/) ) = if ( B = A , C , (/) ) )
27	25, 26	ax-mp 5	. 2 |- if ( A = B , C , (/) ) = if ( B = A , C , (/) )
28	24, 27	syl6eq 2512	1 |- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( B = A , C , (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   {cab 2448   e/ wnel 2634   (/)c0 3743   ifcif 3893   {csn 3980   X. cxp 4851   "cima 4856  tag bj-ctag 31613   Proj bj-cproj 31629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-xp 4859  df-rel 4860  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-bj-sngl 31605  df-bj-tag 31614  df-bj-proj 31630

This theorem is referenced by:  bj-pr1val  31643  bj-pr2val  31657