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Theorem bj-pr22val 37020
Description: Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-pr22val pr2𝐴, 𝐵⦆ = 𝐵

Proof of Theorem bj-pr22val
StepHypRef Expression
1 df-bj-2upl 37012 . . . 4 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr2eq 37017 . . . 4 (⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) → pr2𝐴, 𝐵⦆ = pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)))
31, 2ax-mp 5 . . 3 pr2𝐴, 𝐵⦆ = pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
4 bj-pr2un 37018 . . 3 pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
53, 4eqtri 2765 . 2 pr2𝐴, 𝐵⦆ = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
6 df-bj-1upl 36999 . . . . 5 𝐴⦆ = ({∅} × tag 𝐴)
7 bj-pr2eq 37017 . . . . 5 (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2𝐴⦆ = pr2 ({∅} × tag 𝐴))
86, 7ax-mp 5 . . . 4 pr2𝐴⦆ = pr2 ({∅} × tag 𝐴)
9 bj-pr2val 37019 . . . 4 pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10 1n0 8526 . . . . . 6 1o ≠ ∅
1110nesymi 2998 . . . . 5 ¬ ∅ = 1o
1211iffalsei 4535 . . . 4 if(∅ = 1o, 𝐴, ∅) = ∅
138, 9, 123eqtri 2769 . . 3 pr2𝐴⦆ = ∅
14 bj-pr2val 37019 . . . 4 pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15 eqid 2737 . . . . 5 1o = 1o
1615iftruei 4532 . . . 4 if(1o = 1o, 𝐵, ∅) = 𝐵
1714, 16eqtri 2765 . . 3 pr2 ({1o} × tag 𝐵) = 𝐵
1813, 17uneq12i 4166 . 2 (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19 0un 4396 . 2 (∅ ∪ 𝐵) = 𝐵
205, 18, 193eqtri 2769 1 pr2𝐴, 𝐵⦆ = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cun 3949  c0 4333  ifcif 4525  {csn 4626   × cxp 5683  1oc1o 8499  tag bj-ctag 36975  bj-c1upl 36998  bj-c2uple 37011  pr2 bj-cpr2 37015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-1o 8506  df-bj-sngl 36967  df-bj-tag 36976  df-bj-proj 36992  df-bj-1upl 36999  df-bj-2upl 37012  df-bj-pr2 37016
This theorem is referenced by:  bj-2uplth  37022  bj-2uplex  37023
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