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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.
StepHypRef 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 )
31, 2syl6bir 229 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  =  B  ->  ( ( { B }  X. tag  C ) " { A } )  = tag  C
) )
43imp 429 . . . . . . 7  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( ( { B }  X. tag  C ) " { A } )  = tag 
C )
54eleq2d 2508 . . . . . 6  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( { x }  e.  ( ( { B }  X. tag  C ) " { A } )  <->  { x }  e. tag  C )
)
65abbidv 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 }
96, 7, 83eqtr4g 2498 . . . 4  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( A Proj  ( { B }  X. tag  C
) )  =  C )
109ex 434 . . 3  |-  ( A  e.  V  ->  ( A  =  B  ->  ( A Proj  ( { B }  X. tag  C ) )  =  C ) )
11 noel 3638 . . . . 5  |-  -.  {
x }  e.  (/)
127abeq2i 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 )
1413con3i 135 . . . . . . . . 9  |-  ( -.  A  =  B  ->  -.  A  e.  { B } )
15 df-nel 2607 . . . . . . . . 9  |-  ( A  e/  { B }  <->  -.  A  e.  { B } )
1614, 15sylibr 212 . . . . . . . 8  |-  ( -.  A  =  B  ->  A  e/  { B }
)
17 bj-xpima1sn 32178 . . . . . . . 8  |-  ( A  e/  { B }  ->  ( ( { B }  X. tag  C ) " { A } )  =  (/) )
1816, 17syl 16 . . . . . . 7  |-  ( -.  A  =  B  -> 
( ( { B }  X. tag  C ) " { A } )  =  (/) )
1918eleq2d 2508 . . . . . 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 303 . . . 4  |-  ( -.  A  =  B  ->  -.  x  e.  ( A Proj  ( { B }  X. tag  C ) ) )
2221eq0rdv 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
) )  =  (/) ) ) )
2423biimpri 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 ,  (/) ) )
2510, 22, 24sylancl 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 ,  (/) ) )
2826, 27ax-mp 5 . 2  |-  if ( A  =  B ,  C ,  (/) )  =  if ( B  =  A ,  C ,  (/) )
2925, 28syl6eq 2489 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 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
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