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Theorem bj-pr22val 36407
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 36399 . . . 4 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr2eq 36404 . . . 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 36405 . . 3 pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
53, 4eqtri 2754 . 2 pr2𝐴, 𝐵⦆ = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
6 df-bj-1upl 36386 . . . . 5 𝐴⦆ = ({∅} × tag 𝐴)
7 bj-pr2eq 36404 . . . . 5 (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2𝐴⦆ = pr2 ({∅} × tag 𝐴))
86, 7ax-mp 5 . . . 4 pr2𝐴⦆ = pr2 ({∅} × tag 𝐴)
9 bj-pr2val 36406 . . . 4 pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10 1n0 8489 . . . . . 6 1o ≠ ∅
1110nesymi 2992 . . . . 5 ¬ ∅ = 1o
1211iffalsei 4533 . . . 4 if(∅ = 1o, 𝐴, ∅) = ∅
138, 9, 123eqtri 2758 . . 3 pr2𝐴⦆ = ∅
14 bj-pr2val 36406 . . . 4 pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15 eqid 2726 . . . . 5 1o = 1o
1615iftruei 4530 . . . 4 if(1o = 1o, 𝐵, ∅) = 𝐵
1714, 16eqtri 2754 . . 3 pr2 ({1o} × tag 𝐵) = 𝐵
1813, 17uneq12i 4156 . 2 (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19 0un 4387 . 2 (∅ ∪ 𝐵) = 𝐵
205, 18, 193eqtri 2758 1 pr2𝐴, 𝐵⦆ = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cun 3941  c0 4317  ifcif 4523  {csn 4623   × cxp 5667  1oc1o 8460  tag bj-ctag 36362  bj-c1upl 36385  bj-c2uple 36398  pr2 bj-cpr2 36402
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-suc 6364  df-1o 8467  df-bj-sngl 36354  df-bj-tag 36363  df-bj-proj 36379  df-bj-1upl 36386  df-bj-2upl 36399  df-bj-pr2 36403
This theorem is referenced by:  bj-2uplth  36409  bj-2uplex  36410
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