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Theorem bj-pr1val 34747
 Description: Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr1val pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)

Proof of Theorem bj-pr1val
StepHypRef Expression
1 df-bj-pr1 34744 . 2 pr1 ({𝐴} × tag 𝐵) = (∅ Proj ({𝐴} × tag 𝐵))
2 0ex 5180 . . 3 ∅ ∈ V
3 bj-projval 34739 . . 3 (∅ ∈ V → (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅))
42, 3ax-mp 5 . 2 (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅)
51, 4eqtri 2781 1 pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ∅c0 4227  ifcif 4423  {csn 4525   × cxp 5525  tag bj-ctag 34717   Proj bj-cproj 34733  pr1 bj-cpr1 34743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-bj-sngl 34709  df-bj-tag 34718  df-bj-proj 34734  df-bj-pr1 34744 This theorem is referenced by:  bj-pr11val  34748  bj-pr21val  34756
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