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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.
StepHypRef 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 )
31, 2syl6bir 229 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  =  B  ->  ( ( { B }  X. tag  C ) " { A } )  = tag  C
) )
43imp 427 . . . . . . 7  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( ( { B }  X. tag  C ) " { A } )  = tag 
C )
54eleq2d 2524 . . . . . 6  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( { x }  e.  ( ( { B }  X. tag  C ) " { A } )  <->  { x }  e. tag  C )
)
65abbidv 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 }
96, 7, 83eqtr4g 2520 . . . 4  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( A Proj  ( { B }  X. tag  C
) )  =  C )
109ex 432 . . 3  |-  ( A  e.  V  ->  ( A  =  B  ->  ( A Proj  ( { B }  X. tag  C ) )  =  C ) )
11 noel 3787 . . . . 5  |-  -.  {
x }  e.  (/)
127abeq2i 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 )
1413con3i 135 . . . . . . . . 9  |-  ( -.  A  =  B  ->  -.  A  e.  { B } )
15 df-nel 2652 . . . . . . . . 9  |-  ( A  e/  { B }  <->  -.  A  e.  { B } )
1614, 15sylibr 212 . . . . . . . 8  |-  ( -.  A  =  B  ->  A  e/  { B }
)
17 bj-xpima1sn 34933 . . . . . . . 8  |-  ( A  e/  { B }  ->  ( ( { B }  X. tag  C ) " { A } )  =  (/) )
1816, 17syl 16 . . . . . . 7  |-  ( -.  A  =  B  -> 
( ( { B }  X. tag  C ) " { A } )  =  (/) )
1918eleq2d 2524 . . . . . 6  |-  ( -.  A  =  B  -> 
( { x }  e.  ( ( { B }  X. tag  C ) " { A } )  <->  { x }  e.  (/) ) )
2012, 19syl5bb 257 . . . . 5  |-  ( -.  A  =  B  -> 
( x  e.  ( A Proj  ( { B }  X. tag  C ) )  <->  { x }  e.  (/) ) )
2111, 20mtbiri 301 . . . 4  |-  ( -.  A  =  B  ->  -.  x  e.  ( A Proj  ( { B }  X. tag  C ) ) )
2221eq0rdv 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
) )  =  (/) ) ) )
2410, 22, 23sylancl3 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 ,  (/) ) )
2725, 26ax-mp 5 . 2  |-  if ( A  =  B ,  C ,  (/) )  =  if ( B  =  A ,  C ,  (/) )
2824, 27syl6eq 2511 1  |-  ( A  e.  V  ->  ( A Proj  ( { B }  X. tag  C ) )  =  if ( B  =  A ,  C ,  (/) ) )
Colors of variables: wff setvar class
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
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