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Theorem bj-pr1val 36987
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 36984 . 2 pr1 ({𝐴} × tag 𝐵) = (∅ Proj ({𝐴} × tag 𝐵))
2 0ex 5313 . . 3 ∅ ∈ V
3 bj-projval 36979 . . 3 (∅ ∈ V → (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅))
42, 3ax-mp 5 . 2 (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅)
51, 4eqtri 2763 1 pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  ifcif 4531  {csn 4631   × cxp 5687  tag bj-ctag 36957   Proj bj-cproj 36973  pr1 bj-cpr1 36983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-bj-sngl 36949  df-bj-tag 36958  df-bj-proj 36974  df-bj-pr1 36984
This theorem is referenced by:  bj-pr11val  36988  bj-pr21val  36996
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