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Theorem bj-pr2val 32844
Description: Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr2val  |- pr2  ( { A }  X. tag  B )  =  if ( A  =  1o ,  B ,  (/) )

Proof of Theorem bj-pr2val
StepHypRef Expression
1 df-bj-pr2 32841 . 2  |- pr2  ( { A }  X. tag  B )  =  ( 1o Proj  ( { A }  X. tag  B
) )
2 bj-1ex 32775 . . 3  |-  1o  e.  _V
3 bj-projval 32822 . . 3  |-  ( 1o  e.  _V  ->  ( 1o Proj  ( { A }  X. tag  B ) )  =  if ( A  =  1o ,  B ,  (/) ) )
42, 3ax-mp 5 . 2  |-  ( 1o Proj 
( { A }  X. tag  B ) )  =  if ( A  =  1o ,  B ,  (/) )
51, 4eqtri 2483 1  |- pr2  ( { A }  X. tag  B )  =  if ( A  =  1o ,  B ,  (/) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3746   ifcif 3900   {csn 3986    X. cxp 4947   1oc1o 7024  tag bj-ctag 32800   Proj bj-cproj 32816  pr2 bj-cpr2 32840
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-1o 7031  df-bj-sngl 32792  df-bj-tag 32801  df-bj-proj 32817  df-bj-pr2 32841
This theorem is referenced by:  bj-pr22val  32845
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