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Theorem bj-pr1val 32967
 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 32964 . 2 pr1 ({𝐴} × tag 𝐵) = (∅ Proj ({𝐴} × tag 𝐵))
2 0ex 4781 . . 3 ∅ ∈ V
3 bj-projval 32959 . . 3 (∅ ∈ V → (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅))
42, 3ax-mp 5 . 2 (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅)
51, 4eqtri 2642 1 pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1481   ∈ wcel 1988  Vcvv 3195  ∅c0 3907  ifcif 4077  {csn 4168   × cxp 5102  tag bj-ctag 32937   Proj bj-cproj 32953  pr1 bj-cpr1 32963 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-bj-sngl 32929  df-bj-tag 32938  df-bj-proj 32954  df-bj-pr1 32964 This theorem is referenced by:  bj-pr11val  32968  bj-pr21val  32976
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