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Theorem bj-pr2val 36406
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 36403 . 2 pr2 ({𝐴} × tag 𝐵) = (1o Proj ({𝐴} × tag 𝐵))
2 1oex 8477 . . 3 1o ∈ V
3 bj-projval 36384 . . 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 2754 1 pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3468  c0 4317  ifcif 4523  {csn 4623   × cxp 5667  1oc1o 8460  tag bj-ctag 36362   Proj bj-cproj 36378  pr2 bj-cpr2 36402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-suc 6364  df-1o 8467  df-bj-sngl 36354  df-bj-tag 36363  df-bj-proj 36379  df-bj-pr2 36403
This theorem is referenced by:  bj-pr22val  36407
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