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Theorem bj-pr21val 37015
Description: Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-pr21val pr1𝐴, 𝐵⦆ = 𝐴

Proof of Theorem bj-pr21val
StepHypRef Expression
1 df-bj-2upl 37013 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr1eq 37004 . . 3 (⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) → pr1𝐴, 𝐵⦆ = pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)))
31, 2ax-mp 5 . 2 pr1𝐴, 𝐵⦆ = pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
4 bj-pr1un 37005 . 2 pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5 bj-pr11val 37007 . . . 4 pr1𝐴⦆ = 𝐴
6 bj-pr1val 37006 . . . . 5 pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7 1n0 8527 . . . . . . 7 1o ≠ ∅
87neii 2941 . . . . . 6 ¬ 1o = ∅
98iffalsei 4534 . . . . 5 if(1o = ∅, 𝐵, ∅) = ∅
106, 9eqtri 2764 . . . 4 pr1 ({1o} × tag 𝐵) = ∅
115, 10uneq12i 4165 . . 3 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12 un0 4393 . . 3 (𝐴 ∪ ∅) = 𝐴
1311, 12eqtri 2764 . 2 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
143, 4, 133eqtri 2768 1 pr1𝐴, 𝐵⦆ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cun 3948  c0 4332  ifcif 4524  {csn 4625   × cxp 5682  1oc1o 8500  tag bj-ctag 36976  bj-c1upl 36999  pr1 bj-cpr1 37002  bj-c2uple 37012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-suc 6389  df-1o 8507  df-bj-sngl 36968  df-bj-tag 36977  df-bj-proj 36993  df-bj-1upl 37000  df-bj-pr1 37003  df-bj-2upl 37013
This theorem is referenced by:  bj-2uplth  37023  bj-2uplex  37024
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