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Theorem bj-pr1val 32810
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 32807 . 2  |- pr1  ( { A }  X. tag  B )  =  ( (/) Proj  ( { A }  X. tag  B
) )
2 0ex 4525 . . 3  |-  (/)  e.  _V
3 bj-projval 32802 . . 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 2481 1  |- pr1  ( { A }  X. tag  B )  =  if ( A  =  (/) ,  B ,  (/) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3072   (/)c0 3740   ifcif 3894   {csn 3980    X. cxp 4941  tag bj-ctag 32780   Proj bj-cproj 32796  pr1 bj-cpr1 32806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-xp 4949  df-rel 4950  df-cnv 4951  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-bj-sngl 32772  df-bj-tag 32781  df-bj-proj 32797  df-bj-pr1 32807
This theorem is referenced by:  bj-pr11val  32811  bj-pr21val  32819
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