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Theorem bj-pr1val 33316
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 33313 . 2 pr1 ({𝐴} × tag 𝐵) = (∅ Proj ({𝐴} × tag 𝐵))
2 0ex 4942 . . 3 ∅ ∈ V
3 bj-projval 33308 . . 3 (∅ ∈ V → (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅))
42, 3ax-mp 5 . 2 (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅)
51, 4eqtri 2782 1 pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  ifcif 4230  {csn 4321   × cxp 5264  tag bj-ctag 33286   Proj bj-cproj 33302  pr1 bj-cpr1 33312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-bj-sngl 33278  df-bj-tag 33287  df-bj-proj 33303  df-bj-pr1 33313
This theorem is referenced by:  bj-pr11val  33317  bj-pr21val  33325
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