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Theorem bj-projval 31545
Description: Value of the class projection (proof can be shortened by 19 bytes by using sylancl3 31155). (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 4019 . . . . . . . . 9  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
2 bj-xpima2sn 31506 . . . . . . . . 9  |-  ( A  e.  { B }  ->  ( ( { B }  X. tag  C ) " { A } )  = tag 
C )
31, 2syl6bir 232 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  =  B  ->  ( ( { B }  X. tag  C ) " { A } )  = tag  C
) )
43imp 430 . . . . . . 7  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( ( { B }  X. tag  C ) " { A } )  = tag 
C )
54eleq2d 2492 . . . . . 6  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( { x }  e.  ( ( { B }  X. tag  C ) " { A } )  <->  { x }  e. tag  C )
)
65abbidv 2558 . . . . 5  |-  ( ( A  e.  V  /\  A  =  B )  ->  { x  |  {
x }  e.  ( ( { B }  X. tag  C ) " { A } ) }  =  { x  |  {
x }  e. tag  C } )
7 df-bj-proj 31540 . . . . 5  |-  ( A Proj  ( { B }  X. tag  C ) )  =  { x  |  {
x }  e.  ( ( { B }  X. tag  C ) " { A } ) }
8 bj-taginv 31535 . . . . 5  |-  C  =  { x  |  {
x }  e. tag  C }
96, 7, 83eqtr4g 2488 . . . 4  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( A Proj  ( { B }  X. tag  C
) )  =  C )
109ex 435 . . 3  |-  ( A  e.  V  ->  ( A  =  B  ->  ( A Proj  ( { B }  X. tag  C ) )  =  C ) )
11 noel 3765 . . . . 5  |-  -.  {
x }  e.  (/)
127abeq2i 2549 . . . . . 6  |-  ( x  e.  ( A Proj  ( { B }  X. tag  C
) )  <->  { x }  e.  ( ( { B }  X. tag  C
) " { A } ) )
13 elsni 4021 . . . . . . . . . 10  |-  ( A  e.  { B }  ->  A  =  B )
1413con3i 140 . . . . . . . . 9  |-  ( -.  A  =  B  ->  -.  A  e.  { B } )
15 df-nel 2621 . . . . . . . . 9  |-  ( A  e/  { B }  <->  -.  A  e.  { B } )
1614, 15sylibr 215 . . . . . . . 8  |-  ( -.  A  =  B  ->  A  e/  { B }
)
17 bj-xpima1sn 31504 . . . . . . . 8  |-  ( A  e/  { B }  ->  ( ( { B }  X. tag  C ) " { A } )  =  (/) )
1816, 17syl 17 . . . . . . 7  |-  ( -.  A  =  B  -> 
( ( { B }  X. tag  C ) " { A } )  =  (/) )
1918eleq2d 2492 . . . . . 6  |-  ( -.  A  =  B  -> 
( { x }  e.  ( ( { B }  X. tag  C ) " { A } )  <->  { x }  e.  (/) ) )
2012, 19syl5bb 260 . . . . 5  |-  ( -.  A  =  B  -> 
( x  e.  ( A Proj  ( { B }  X. tag  C ) )  <->  { x }  e.  (/) ) )
2111, 20mtbiri 304 . . . 4  |-  ( -.  A  =  B  ->  -.  x  e.  ( A Proj  ( { B }  X. tag  C ) ) )
2221eq0rdv 3797 . . 3  |-  ( -.  A  =  B  -> 
( A Proj  ( { B }  X. tag  C ) )  =  (/) )
23 ifval 3948 . . 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 31155 . 2  |-  ( A  e.  V  ->  ( A Proj  ( { B }  X. tag  C ) )  =  if ( A  =  B ,  C ,  (/) ) )
25 eqcom 2431 . . 3  |-  ( A  =  B  <->  B  =  A )
26 ifbi 3930 . . 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 2479 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 187    /\ wa 370    = wceq 1437    e. wcel 1868   {cab 2407    e/ wnel 2619   (/)c0 3761   ifcif 3909   {csn 3996    X. cxp 4847   "cima 4852  tag bj-ctag 31523   Proj bj-cproj 31539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-xp 4855  df-rel 4856  df-cnv 4857  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-bj-sngl 31515  df-bj-tag 31524  df-bj-proj 31540
This theorem is referenced by:  bj-pr1val  31553  bj-pr2val  31567
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