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Theorem bj-pr21val 36625
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 36623 . . 3 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr1eq 36614 . . 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 36615 . 2 pr1 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵))
5 bj-pr11val 36617 . . . 4 pr1𝐴⦆ = 𝐴
6 bj-pr1val 36616 . . . . 5 pr1 ({1o} × tag 𝐵) = if(1o = ∅, 𝐵, ∅)
7 1n0 8509 . . . . . . 7 1o ≠ ∅
87neii 2931 . . . . . 6 ¬ 1o = ∅
98iffalsei 4540 . . . . 5 if(1o = ∅, 𝐵, ∅) = ∅
106, 9eqtri 2753 . . . 4 pr1 ({1o} × tag 𝐵) = ∅
115, 10uneq12i 4158 . . 3 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = (𝐴 ∪ ∅)
12 un0 4392 . . 3 (𝐴 ∪ ∅) = 𝐴
1311, 12eqtri 2753 . 2 (pr1𝐴⦆ ∪ pr1 ({1o} × tag 𝐵)) = 𝐴
143, 4, 133eqtri 2757 1 pr1𝐴, 𝐵⦆ = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cun 3942  c0 4322  ifcif 4530  {csn 4630   × cxp 5676  1oc1o 8480  tag bj-ctag 36586  bj-c1upl 36609  pr1 bj-cpr1 36612  bj-c2uple 36622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-suc 6377  df-1o 8487  df-bj-sngl 36578  df-bj-tag 36587  df-bj-proj 36603  df-bj-1upl 36610  df-bj-pr1 36613  df-bj-2upl 36623
This theorem is referenced by:  bj-2uplth  36633  bj-2uplex  36634
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