Theorem bj-projval 34974

Description: Value of the class projection (proof can be shortened by 19 bytes by using sylancl3 34589). (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 4039	. . . . . . . . 9 |- ( A e. V -> ( A e. { B } <-> A = B ) )
2		bj-xpima2sn 34935	. . . . . . . . 9 |- ( A e. { B } -> ( ( { B } X. tag C ) " { A } ) = tag C )
3	1, 2	syl6bir 229	. . . . . . . 8 |- ( A e. V -> ( A = B -> ( ( { B } X. tag C ) " { A } ) = tag C ) )
4	3	imp 427	. . . . . . 7 |- ( ( A e. V /\ A = B ) -> ( ( { B } X. tag C ) " { A } ) = tag C )
5	4	eleq2d 2524	. . . . . 6 |- ( ( A e. V /\ A = B ) -> ( { x } e. ( ( { B } X. tag C ) " { A } ) <-> { x } e. tag C ) )
6	5	abbidv 2590	. . . . 5 |- ( ( A e. V /\ A = B ) -> { x | { x } e. ( ( { B } X. tag C ) " { A } ) } = { x | { x } e. tag C } )
7		df-bj-proj 34969	. . . . 5 |- ( A Proj ( { B } X. tag C ) ) = { x | { x } e. ( ( { B } X. tag C ) " { A } ) }
8		bj-taginv 34964	. . . . 5 |- C = { x | { x } e. tag C }
9	6, 7, 8	3eqtr4g 2520	. . . 4 |- ( ( A e. V /\ A = B ) -> ( A Proj ( { B } X. tag C ) ) = C )
10	9	ex 432	. . 3 |- ( A e. V -> ( A = B -> ( A Proj ( { B } X. tag C ) ) = C ) )
11		noel 3787	. . . . 5 |- -. { x } e. (/)
12	7	abeq2i 2581	. . . . . 6 |- ( x e. ( A Proj ( { B } X. tag C ) ) <-> { x } e. ( ( { B } X. tag C ) " { A } ) )
13		elsni 4041	. . . . . . . . . 10 |- ( A e. { B } -> A = B )
14	13	con3i 135	. . . . . . . . 9 |- ( -. A = B -> -. A e. { B } )
15		df-nel 2652	. . . . . . . . 9 |- ( A e/ { B } <-> -. A e. { B } )
16	14, 15	sylibr 212	. . . . . . . 8 |- ( -. A = B -> A e/ { B } )
17		bj-xpima1sn 34933	. . . . . . . 8 |- ( A e/ { B } -> ( ( { B } X. tag C ) " { A } ) = (/) )
18	16, 17	syl 16	. . . . . . 7 |- ( -. A = B -> ( ( { B } X. tag C ) " { A } ) = (/) )
19	18	eleq2d 2524	. . . . . 6 |- ( -. A = B -> ( { x } e. ( ( { B } X. tag C ) " { A } ) <-> { x } e. (/) ) )
20	12, 19	syl5bb 257	. . . . 5 |- ( -. A = B -> ( x e. ( A Proj ( { B } X. tag C ) ) <-> { x } e. (/) ) )
21	11, 20	mtbiri 301	. . . 4 |- ( -. A = B -> -. x e. ( A Proj ( { B } X. tag C ) ) )
22	21	eq0rdv 3819	. . 3 |- ( -. A = B -> ( A Proj ( { B } X. tag C ) ) = (/) )
23		ifval 3968	. . 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 34589	. 2 |- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( A = B , C , (/) ) )
25		eqcom 2463	. . 3 |- ( A = B <-> B = A )
26		ifbi 3950	. . 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 2511	1 |- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( B = A , C , (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   {cab 2439   e/ wnel 2650   (/)c0 3783   ifcif 3929   {csn 4016   X. cxp 4986   "cima 4991  tag bj-ctag 34952   Proj bj-cproj 34968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-bj-sngl 34944  df-bj-tag 34953  df-bj-proj 34969

This theorem is referenced by:  bj-pr1val  34982  bj-pr2val  34996