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Theorem bj-pr2val 35804
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 35801 . 2 pr2 ({𝐴} × tag 𝐵) = (1o Proj ({𝐴} × tag 𝐵))
2 1oex 8463 . . 3 1o ∈ V
3 bj-projval 35782 . . 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 2761 1 pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3475  c0 4320  ifcif 4524  {csn 4624   × cxp 5670  1oc1o 8446  tag bj-ctag 35760   Proj bj-cproj 35776  pr2 bj-cpr2 35800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-suc 6362  df-1o 8453  df-bj-sngl 35752  df-bj-tag 35761  df-bj-proj 35777  df-bj-pr2 35801
This theorem is referenced by:  bj-pr22val  35805
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