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Theorem bj-pr2val 35894
Description: Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr2val pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)

Proof of Theorem bj-pr2val
StepHypRef Expression
1 df-bj-pr2 35891 . 2 pr2 ({𝐴} × tag 𝐵) = (1o Proj ({𝐴} × tag 𝐵))
2 1oex 8475 . . 3 1o ∈ V
3 bj-projval 35872 . . 3 (1o ∈ V → (1o Proj ({𝐴} × tag 𝐵)) = if(𝐴 = 1o, 𝐵, ∅))
42, 3ax-mp 5 . 2 (1o Proj ({𝐴} × tag 𝐵)) = if(𝐴 = 1o, 𝐵, ∅)
51, 4eqtri 2760 1 pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  Vcvv 3474  c0 4322  ifcif 4528  {csn 4628   × cxp 5674  1oc1o 8458  tag bj-ctag 35850   Proj bj-cproj 35866  pr2 bj-cpr2 35890
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-1o 8465  df-bj-sngl 35842  df-bj-tag 35851  df-bj-proj 35867  df-bj-pr2 35891
This theorem is referenced by:  bj-pr22val  35895
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