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Theorem bj-pr21val 36996
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 36994 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr1eq 36985 . . 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 36986 . 2 pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5 bj-pr11val 36988 . . . 4 pr1𝐴⦆ = 𝐴
6 bj-pr1val 36987 . . . . 5 pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7 1n0 8525 . . . . . . 7 1o ≠ ∅
87neii 2940 . . . . . 6 ¬ 1o = ∅
98iffalsei 4541 . . . . 5 if(1o = ∅, 𝐵, ∅) = ∅
106, 9eqtri 2763 . . . 4 pr1 ({1o} × tag 𝐵) = ∅
115, 10uneq12i 4176 . . 3 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12 un0 4400 . . 3 (𝐴 ∪ ∅) = 𝐴
1311, 12eqtri 2763 . 2 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
143, 4, 133eqtri 2767 1 pr1𝐴, 𝐵⦆ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cun 3961  c0 4339  ifcif 4531  {csn 4631   × cxp 5687  1oc1o 8498  tag bj-ctag 36957  bj-c1upl 36980  pr1 bj-cpr1 36983  bj-c2uple 36993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-1o 8505  df-bj-sngl 36949  df-bj-tag 36958  df-bj-proj 36974  df-bj-1upl 36981  df-bj-pr1 36984  df-bj-2upl 36994
This theorem is referenced by:  bj-2uplth  37004  bj-2uplex  37005
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