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Theorem bj-pr21val 33872
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 33870 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr1eq 33861 . . 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 33862 . 2 pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5 bj-pr11val 33864 . . . 4 pr1𝐴⦆ = 𝐴
6 bj-pr1val 33863 . . . . 5 pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7 1n0 7919 . . . . . . 7 1o ≠ ∅
87neii 2963 . . . . . 6 ¬ 1o = ∅
98iffalsei 4354 . . . . 5 if(1o = ∅, 𝐵, ∅) = ∅
106, 9eqtri 2796 . . . 4 pr1 ({1o} × tag 𝐵) = ∅
115, 10uneq12i 4020 . . 3 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12 un0 4224 . . 3 (𝐴 ∪ ∅) = 𝐴
1311, 12eqtri 2796 . 2 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
143, 4, 133eqtri 2800 1 pr1𝐴, 𝐵⦆ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1507  cun 3821  c0 4172  ifcif 4344  {csn 4435   × cxp 5401  1oc1o 7896  tag bj-ctag 33833  bj-c1upl 33856  pr1 bj-cpr1 33859  bj-c2uple 33869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-xp 5409  df-rel 5410  df-cnv 5411  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-suc 6032  df-1o 7903  df-bj-sngl 33825  df-bj-tag 33834  df-bj-proj 33850  df-bj-1upl 33857  df-bj-pr1 33860  df-bj-2upl 33870
This theorem is referenced by:  bj-2uplth  33880  bj-2uplex  33881
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