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Theorem bj-pr2val 34963
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 34960 . 2  |- pr2  ( { A }  X. tag  B )  =  ( 1o Proj  ( { A }  X. tag  B
) )
2 bj-1ex 34894 . . 3  |-  1o  e.  _V
3 bj-projval 34941 . . 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 2425 1  |- pr2  ( { A }  X. tag  B )  =  if ( A  =  1o ,  B ,  (/) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    e. wcel 1836   _Vcvv 3051   (/)c0 3728   ifcif 3874   {csn 3961    X. cxp 4928   1oc1o 7063  tag bj-ctag 34919   Proj bj-cproj 34935  pr2 bj-cpr2 34959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-1o 7070  df-bj-sngl 34911  df-bj-tag 34920  df-bj-proj 34936  df-bj-pr2 34960
This theorem is referenced by:  bj-pr22val  34964
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