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Theorem bj-pr1val 37006
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 37003 . 2 pr1 ({𝐴} × tag 𝐵) = (∅ Proj ({𝐴} × tag 𝐵))
2 0ex 5306 . . 3 ∅ ∈ V
3 bj-projval 36998 . . 3 (∅ ∈ V → (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅))
42, 3ax-mp 5 . 2 (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅)
51, 4eqtri 2764 1 pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  Vcvv 3479  c0 4332  ifcif 4524  {csn 4625   × cxp 5682  tag bj-ctag 36976   Proj bj-cproj 36992  pr1 bj-cpr1 37002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-bj-sngl 36968  df-bj-tag 36977  df-bj-proj 36993  df-bj-pr1 37003
This theorem is referenced by:  bj-pr11val  37007  bj-pr21val  37015
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