Theorem bj-projval 32984

Description: Value of the class projection. (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		elsng 4191	. . . . . . . . 9 |- ( A e. V -> ( A e. { B } <-> A = B ) )
2		bj-xpima2sn 32945	. . . . . . . . 9 |- ( A e. { B } -> ( ( { B } X. tag C ) " { A } ) = tag C )
3	1, 2	syl6bir 244	. . . . . . . 8 |- ( A e. V -> ( A = B -> ( ( { B } X. tag C ) " { A } ) = tag C ) )
4	3	imp 445	. . . . . . 7 |- ( ( A e. V /\ A = B ) -> ( ( { B } X. tag C ) " { A } ) = tag C )
5	4	eleq2d 2687	. . . . . 6 |- ( ( A e. V /\ A = B ) -> ( { x } e. ( ( { B } X. tag C ) " { A } ) <-> { x } e. tag C ) )
6	5	abbidv 2741	. . . . 5 |- ( ( A e. V /\ A = B ) -> { x | { x } e. ( ( { B } X. tag C ) " { A } ) } = { x | { x } e. tag C } )
7		df-bj-proj 32979	. . . . 5 |- ( A Proj ( { B } X. tag C ) ) = { x | { x } e. ( ( { B } X. tag C ) " { A } ) }
8		bj-taginv 32974	. . . . 5 |- C = { x | { x } e. tag C }
9	6, 7, 8	3eqtr4g 2681	. . . 4 |- ( ( A e. V /\ A = B ) -> ( A Proj ( { B } X. tag C ) ) = C )
10	9	ex 450	. . 3 |- ( A e. V -> ( A = B -> ( A Proj ( { B } X. tag C ) ) = C ) )
11		noel 3919	. . . . 5 |- -. { x } e. (/)
12	7	abeq2i 2735	. . . . . 6 |- ( x e. ( A Proj ( { B } X. tag C ) ) <-> { x } e. ( ( { B } X. tag C ) " { A } ) )
13		elsni 4194	. . . . . . . . . 10 |- ( A e. { B } -> A = B )
14	13	con3i 150	. . . . . . . . 9 |- ( -. A = B -> -. A e. { B } )
15		df-nel 2898	. . . . . . . . 9 |- ( A e/ { B } <-> -. A e. { B } )
16	14, 15	sylibr 224	. . . . . . . 8 |- ( -. A = B -> A e/ { B } )
17		bj-xpima1sn 32943	. . . . . . . 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 2687	. . . . . 6 |- ( -. A = B -> ( { x } e. ( ( { B } X. tag C ) " { A } ) <-> { x } e. (/) ) )
20	12, 19	syl5bb 272	. . . . 5 |- ( -. A = B -> ( x e. ( A Proj ( { B } X. tag C ) ) <-> { x } e. (/) ) )
21	11, 20	mtbiri 317	. . . 4 |- ( -. A = B -> -. x e. ( A Proj ( { B } X. tag C ) ) )
22	21	eq0rdv 3979	. . 3 |- ( -. A = B -> ( A Proj ( { B } X. tag C ) ) = (/) )
23		ifval 4127	. . 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	sylanblrc 697	. 2 |- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( A = B , C , (/) ) )
25		eqcom 2629	. . 3 |- ( A = B <-> B = A )
26		ifbi 4107	. . 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 2672	1 |- ( A e. V -> ( A Proj ( { B } X. tag C ) ) = if ( B = A , C , (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   e/ wnel 2897   (/)c0 3915   ifcif 4086   {csn 4177   X. cxp 5112   "cima 5117  tag bj-ctag 32962   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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  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-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-bj-sngl 32954  df-bj-tag 32963  df-bj-proj 32979

This theorem is referenced by:  bj-pr1val  32992  bj-pr2val  33006