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Theorem bj-pr1val 34963
Description: Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr1val  |- pr1  ( { A }  X. tag  B )  =  if ( A  =  (/) ,  B ,  (/) )

Proof of Theorem bj-pr1val
StepHypRef Expression
1 df-bj-pr1 34960 . 2  |- pr1  ( { A }  X. tag  B )  =  ( (/) Proj  ( { A }  X. tag  B
) )
2 0ex 4569 . . 3  |-  (/)  e.  _V
3 bj-projval 34955 . . 3  |-  ( (/)  e.  _V  ->  ( (/) Proj  ( { A }  X. tag  B
) )  =  if ( A  =  (/) ,  B ,  (/) ) )
42, 3ax-mp 5 . 2  |-  ( (/) Proj  ( { A }  X. tag  B ) )  =  if ( A  =  (/) ,  B ,  (/) )
51, 4eqtri 2483 1  |- pr1  ( { A }  X. tag  B )  =  if ( A  =  (/) ,  B ,  (/) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   ifcif 3929   {csn 4016    X. cxp 4986  tag bj-ctag 34933   Proj bj-cproj 34949  pr1 bj-cpr1 34959
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 34925  df-bj-tag 34934  df-bj-proj 34950  df-bj-pr1 34960
This theorem is referenced by:  bj-pr11val  34964  bj-pr21val  34972
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