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Theorem bj-projval 31635
Description: Value of the class projection (proof can be shortened by 19 bytes by using sylancl3 31209). (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 4003 . . . . . . . . 9  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
2 bj-xpima2sn 31596 . . . . . . . . 9  |-  ( A  e.  { B }  ->  ( ( { B }  X. tag  C ) " { A } )  = tag 
C )
31, 2syl6bir 237 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  =  B  ->  ( ( { B }  X. tag  C ) " { A } )  = tag  C
) )
43imp 435 . . . . . . 7  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( ( { B }  X. tag  C ) " { A } )  = tag 
C )
54eleq2d 2525 . . . . . 6  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( { x }  e.  ( ( { B }  X. tag  C ) " { A } )  <->  { x }  e. tag  C )
)
65abbidv 2580 . . . . 5  |-  ( ( A  e.  V  /\  A  =  B )  ->  { x  |  {
x }  e.  ( ( { B }  X. tag  C ) " { A } ) }  =  { x  |  {
x }  e. tag  C } )
7 df-bj-proj 31630 . . . . 5  |-  ( A Proj  ( { B }  X. tag  C ) )  =  { x  |  {
x }  e.  ( ( { B }  X. tag  C ) " { A } ) }
8 bj-taginv 31625 . . . . 5  |-  C  =  { x  |  {
x }  e. tag  C }
96, 7, 83eqtr4g 2521 . . . 4  |-  ( ( A  e.  V  /\  A  =  B )  ->  ( A Proj  ( { B }  X. tag  C
) )  =  C )
109ex 440 . . 3  |-  ( A  e.  V  ->  ( A  =  B  ->  ( A Proj  ( { B }  X. tag  C ) )  =  C ) )
11 noel 3747 . . . . 5  |-  -.  {
x }  e.  (/)
127abeq2i 2574 . . . . . 6  |-  ( x  e.  ( A Proj  ( { B }  X. tag  C
) )  <->  { x }  e.  ( ( { B }  X. tag  C
) " { A } ) )
13 elsni 4005 . . . . . . . . . 10  |-  ( A  e.  { B }  ->  A  =  B )
1413con3i 142 . . . . . . . . 9  |-  ( -.  A  =  B  ->  -.  A  e.  { B } )
15 df-nel 2636 . . . . . . . . 9  |-  ( A  e/  { B }  <->  -.  A  e.  { B } )
1614, 15sylibr 217 . . . . . . . 8  |-  ( -.  A  =  B  ->  A  e/  { B }
)
17 bj-xpima1sn 31594 . . . . . . . 8  |-  ( A  e/  { B }  ->  ( ( { B }  X. tag  C ) " { A } )  =  (/) )
1816, 17syl 17 . . . . . . 7  |-  ( -.  A  =  B  -> 
( ( { B }  X. tag  C ) " { A } )  =  (/) )
1918eleq2d 2525 . . . . . 6  |-  ( -.  A  =  B  -> 
( { x }  e.  ( ( { B }  X. tag  C ) " { A } )  <->  { x }  e.  (/) ) )
2012, 19syl5bb 265 . . . . 5  |-  ( -.  A  =  B  -> 
( x  e.  ( A Proj  ( { B }  X. tag  C ) )  <->  { x }  e.  (/) ) )
2111, 20mtbiri 309 . . . 4  |-  ( -.  A  =  B  ->  -.  x  e.  ( A Proj  ( { B }  X. tag  C ) ) )
2221eq0rdv 3781 . . 3  |-  ( -.  A  =  B  -> 
( A Proj  ( { B }  X. tag  C ) )  =  (/) )
23 ifval 3932 . . 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 31209 . 2  |-  ( A  e.  V  ->  ( A Proj  ( { B }  X. tag  C ) )  =  if ( A  =  B ,  C ,  (/) ) )
25 eqcom 2469 . . 3  |-  ( A  =  B  <->  B  =  A )
26 ifbi 3914 . . 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 2512 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 189    /\ wa 375    = wceq 1455    e. wcel 1898   {cab 2448    e/ wnel 2634   (/)c0 3743   ifcif 3893   {csn 3980    X. cxp 4851   "cima 4856  tag bj-ctag 31613   Proj bj-cproj 31629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-xp 4859  df-rel 4860  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-bj-sngl 31605  df-bj-tag 31614  df-bj-proj 31630
This theorem is referenced by:  bj-pr1val  31643  bj-pr2val  31657
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