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Theorem bj-pr22val 37002
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 36994 . . . 4 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr2eq 36999 . . . 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 37000 . . 3 pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
53, 4eqtri 2763 . 2 pr2𝐴, 𝐵⦆ = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
6 df-bj-1upl 36981 . . . . 5 𝐴⦆ = ({∅} × tag 𝐴)
7 bj-pr2eq 36999 . . . . 5 (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2𝐴⦆ = pr2 ({∅} × tag 𝐴))
86, 7ax-mp 5 . . . 4 pr2𝐴⦆ = pr2 ({∅} × tag 𝐴)
9 bj-pr2val 37001 . . . 4 pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10 1n0 8525 . . . . . 6 1o ≠ ∅
1110nesymi 2996 . . . . 5 ¬ ∅ = 1o
1211iffalsei 4541 . . . 4 if(∅ = 1o, 𝐴, ∅) = ∅
138, 9, 123eqtri 2767 . . 3 pr2𝐴⦆ = ∅
14 bj-pr2val 37001 . . . 4 pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15 eqid 2735 . . . . 5 1o = 1o
1615iftruei 4538 . . . 4 if(1o = 1o, 𝐵, ∅) = 𝐵
1714, 16eqtri 2763 . . 3 pr2 ({1o} × tag 𝐵) = 𝐵
1813, 17uneq12i 4176 . 2 (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19 0un 4402 . 2 (∅ ∪ 𝐵) = 𝐵
205, 18, 193eqtri 2767 1 pr2𝐴, 𝐵⦆ = 𝐵
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  bj-c2uple 36993  pr2 bj-cpr2 36997
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-2upl 36994  df-bj-pr2 36998
This theorem is referenced by:  bj-2uplth  37004  bj-2uplex  37005
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