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Theorem bj-pr22val 34228
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 34220 . . . 4 𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
2 bj-pr2eq 34225 . . . 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 34226 . . 3 pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
53, 4eqtri 2841 . 2 pr2𝐴, 𝐵⦆ = (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵))
6 df-bj-1upl 34207 . . . . 5 𝐴⦆ = ({∅} × tag 𝐴)
7 bj-pr2eq 34225 . . . . 5 (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2𝐴⦆ = pr2 ({∅} × tag 𝐴))
86, 7ax-mp 5 . . . 4 pr2𝐴⦆ = pr2 ({∅} × tag 𝐴)
9 bj-pr2val 34227 . . . 4 pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅)
10 1n0 8108 . . . . . 6 1o ≠ ∅
1110nesymi 3070 . . . . 5 ¬ ∅ = 1o
1211iffalsei 4473 . . . 4 if(∅ = 1o, 𝐴, ∅) = ∅
138, 9, 123eqtri 2845 . . 3 pr2𝐴⦆ = ∅
14 bj-pr2val 34227 . . . 4 pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅)
15 eqid 2818 . . . . 5 1o = 1o
1615iftruei 4470 . . . 4 if(1o = 1o, 𝐵, ∅) = 𝐵
1714, 16eqtri 2841 . . 3 pr2 ({1o} × tag 𝐵) = 𝐵
1813, 17uneq12i 4134 . 2 (pr2𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵)
19 0un 4343 . 2 (∅ ∪ 𝐵) = 𝐵
205, 18, 193eqtri 2845 1 pr2𝐴, 𝐵⦆ = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cun 3931  c0 4288  ifcif 4463  {csn 4557   × cxp 5546  1oc1o 8084  tag bj-ctag 34183  bj-c1upl 34206  bj-c2uple 34219  pr2 bj-cpr2 34223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-suc 6190  df-1o 8091  df-bj-sngl 34175  df-bj-tag 34184  df-bj-proj 34200  df-bj-1upl 34207  df-bj-2upl 34220  df-bj-pr2 34224
This theorem is referenced by:  bj-2uplth  34230  bj-2uplex  34231
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