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Theorem bj-pr2val 32199
 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 32196 . 2 pr2 ({𝐴} × tag 𝐵) = (1𝑜 Proj ({𝐴} × tag 𝐵))
2 bj-1ex 32131 . . 3 1𝑜 ∈ V
3 bj-projval 32177 . . 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 2632 1 pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1𝑜, 𝐵, ∅)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ifcif 4036  {csn 4125   × cxp 5036  1𝑜c1o 7440  tag bj-ctag 32155   Proj bj-cproj 32171  pr2 bj-cpr2 32195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-1o 7447  df-bj-sngl 32147  df-bj-tag 32156  df-bj-proj 32172  df-bj-pr2 32196 This theorem is referenced by:  bj-pr22val  32200
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