Theorem bj-projval 32219

Description: Value of the class projection (proof can be shortened by 19 bytes by using sylancl3 31906). (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 3897	. . . . . . . . 9 |- ( A e. V -> ( A e. { B } <-> A = B ) )
2		bj-xpima2sn 32180	. . . . . . . . 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 2508	. . . . . 6 |- ( ( A e. V /\ A = B ) -> ( { x } e. ( ( { B } X. tag C ) " { A } ) <-> { x } e. tag C ) )
6	5	abbidv 2555	. . . . 5 |- ( ( A e. V /\ A = B ) -> { x | { x } e. ( ( { B } X. tag C ) " { A } ) } = { x | { x } e. tag C } )
7		df-bj-proj 32214	. . . . 5 |- ( A Proj ( { B } X. tag C ) ) = { x | { x } e. ( ( { B } X. tag C ) " { A } ) }
8		bj-taginv 32209	. . . . 5 |- C = { x | { x } e. tag C }
9	6, 7, 8	3eqtr4g 2498	. . . 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 3638	. . . . 5 |- -. { x } e. (/)
12	7	abeq2i 2548	. . . . . 6 |- ( x e. ( A Proj ( { B } X. tag C ) ) <-> { x } e. ( ( { B } X. tag C ) " { A } ) )
13		elsni 3899	. . . . . . . . . 10 |- ( A e. { B } -> A = B )
14	13	con3i 135	. . . . . . . . 9 |- ( -. A = B -> -. A e. { B } )
15		df-nel 2607	. . . . . . . . 9 |- ( A e/ { B } <-> -. A e. { B } )
16	14, 15	sylibr 212	. . . . . . . 8 |- ( -. A = B -> A e/ { B } )
17		bj-xpima1sn 32178	. . . . . . . 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 2508	. . . . . 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 3669	. . 3 |- ( -. A = B -> ( A Proj ( { B } X. tag C ) ) = (/) )
23		ifval 3825	. . . 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 657	. 2 |- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( A = B , C , (/) ) )
26		eqcom 2443	. . 3 |- ( A = B <-> B = A )
27		ifbi 3807	. . 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 2489	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 1364   e. wcel 1761   {cab 2427   e/ wnel 2605   (/)c0 3634   ifcif 3788   {csn 3874   X. cxp 4834   "cima 4839  tag bj-ctag 32197   Proj bj-cproj 32213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-rel 4843  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-bj-sngl 32189  df-bj-tag 32198  df-bj-proj 32214

This theorem is referenced by:  bj-pr1val  32227  bj-pr2val  32241