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Theorem bj-pr2val 33954
 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 33951 . 2 pr2 ({𝐴} × tag 𝐵) = (1o Proj ({𝐴} × tag 𝐵))
2 1oex 7961 . . 3 1o ∈ V
3 bj-projval 33932 . . 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 2819 1 pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1522   ∈ wcel 2081  Vcvv 3437  ∅c0 4211  ifcif 4381  {csn 4472   × cxp 5441  1oc1o 7946  tag bj-ctag 33910   Proj bj-cproj 33926  pr2 bj-cpr2 33950 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-tr 5064  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-suc 6072  df-1o 7953  df-bj-sngl 33902  df-bj-tag 33911  df-bj-proj 33927  df-bj-pr2 33951 This theorem is referenced by:  bj-pr22val  33955
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