Theorem bj-projval 32494

Description: Value of the class projection (proof can be shortened by 19 bytes by using sylancl3 32111). (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 3905	. . . . . . . . 9 |- ( A e. V -> ( A e. { B } <-> A = B ) )
2		bj-xpima2sn 32455	. . . . . . . . 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 429	. . . . . . 7 |- ( ( A e. V /\ A = B ) -> ( ( { B } X. tag C ) " { A } ) = tag C )
5	4	eleq2d 2510	. . . . . 6 |- ( ( A e. V /\ A = B ) -> ( { x } e. ( ( { B } X. tag C ) " { A } ) <-> { x } e. tag C ) )
6	5	abbidv 2562	. . . . 5 |- ( ( A e. V /\ A = B ) -> { x | { x } e. ( ( { B } X. tag C ) " { A } ) } = { x | { x } e. tag C } )
7		df-bj-proj 32489	. . . . 5 |- ( A Proj ( { B } X. tag C ) ) = { x | { x } e. ( ( { B } X. tag C ) " { A } ) }
8		bj-taginv 32484	. . . . 5 |- C = { x | { x } e. tag C }
9	6, 7, 8	3eqtr4g 2500	. . . 4 |- ( ( A e. V /\ A = B ) -> ( A Proj ( { B } X. tag C ) ) = C )
10	9	ex 434	. . 3 |- ( A e. V -> ( A = B -> ( A Proj ( { B } X. tag C ) ) = C ) )
11		noel 3646	. . . . 5 |- -. { x } e. (/)
12	7	abeq2i 2554	. . . . . 6 |- ( x e. ( A Proj ( { B } X. tag C ) ) <-> { x } e. ( ( { B } X. tag C ) " { A } ) )
13		elsni 3907	. . . . . . . . . 10 |- ( A e. { B } -> A = B )
14	13	con3i 135	. . . . . . . . 9 |- ( -. A = B -> -. A e. { B } )
15		df-nel 2614	. . . . . . . . 9 |- ( A e/ { B } <-> -. A e. { B } )
16	14, 15	sylibr 212	. . . . . . . 8 |- ( -. A = B -> A e/ { B } )
17		bj-xpima1sn 32453	. . . . . . . 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 2510	. . . . . 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 303	. . . 4 |- ( -. A = B -> -. x e. ( A Proj ( { B } X. tag C ) ) )
22	21	eq0rdv 3677	. . 3 |- ( -. A = B -> ( A Proj ( { B } X. tag C ) ) = (/) )
23		ifval 3833	. . . 4 |- ( ( 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	23	biimpri 206	. . 3 |- ( ( ( A = B -> ( A Proj ( { B } X. tag C ) ) = C ) /\ ( -. A = B -> ( A Proj ( { B } X. tag C ) ) = (/) ) ) -> ( A Proj ( { B } X. tag C ) ) = if ( A = B , C , (/) ) )
25	10, 22, 24	sylancl 662	. 2 |- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( A = B , C , (/) ) )
26		eqcom 2445	. . 3 |- ( A = B <-> B = A )
27		ifbi 3815	. . 3 |- ( ( A = B <-> B = A ) -> if ( A = B , C , (/) ) = if ( B = A , C , (/) ) )
28	26, 27	ax-mp 5	. 2 |- if ( A = B , C , (/) ) = if ( B = A , C , (/) )
29	25, 28	syl6eq 2491	1 |- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( B = A , C , (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   {cab 2429   e/ wnel 2612   (/)c0 3642   ifcif 3796   {csn 3882   X. cxp 4843   "cima 4848  tag bj-ctag 32472   Proj bj-cproj 32488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-bj-sngl 32464  df-bj-tag 32473  df-bj-proj 32489

This theorem is referenced by:  bj-pr1val  32502  bj-pr2val  32516