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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr22val | Structured version Visualization version GIF version |
Description: Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-pr22val | ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-2upl 35887 | . . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-pr2eq 35892 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) → pr2 ⦅𝐴, 𝐵⦆ = pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ pr2 ⦅𝐴, 𝐵⦆ = pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) |
4 | bj-pr2un 35893 | . . 3 ⊢ pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) | |
5 | 3, 4 | eqtri 2760 | . 2 ⊢ pr2 ⦅𝐴, 𝐵⦆ = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) |
6 | df-bj-1upl 35874 | . . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
7 | bj-pr2eq 35892 | . . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴) |
9 | bj-pr2val 35894 | . . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅) | |
10 | 1n0 8487 | . . . . . 6 ⊢ 1o ≠ ∅ | |
11 | 10 | nesymi 2998 | . . . . 5 ⊢ ¬ ∅ = 1o |
12 | 11 | iffalsei 4538 | . . . 4 ⊢ if(∅ = 1o, 𝐴, ∅) = ∅ |
13 | 8, 9, 12 | 3eqtri 2764 | . . 3 ⊢ pr2 ⦅𝐴⦆ = ∅ |
14 | bj-pr2val 35894 | . . . 4 ⊢ pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅) | |
15 | eqid 2732 | . . . . 5 ⊢ 1o = 1o | |
16 | 15 | iftruei 4535 | . . . 4 ⊢ if(1o = 1o, 𝐵, ∅) = 𝐵 |
17 | 14, 16 | eqtri 2760 | . . 3 ⊢ pr2 ({1o} × tag 𝐵) = 𝐵 |
18 | 13, 17 | uneq12i 4161 | . 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵) |
19 | 0un 4392 | . 2 ⊢ (∅ ∪ 𝐵) = 𝐵 | |
20 | 5, 18, 19 | 3eqtri 2764 | 1 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∪ cun 3946 ∅c0 4322 ifcif 4528 {csn 4628 × cxp 5674 1oc1o 8458 tag bj-ctag 35850 ⦅bj-c1upl 35873 ⦅bj-c2uple 35886 pr2 bj-cpr2 35890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-suc 6370 df-1o 8465 df-bj-sngl 35842 df-bj-tag 35851 df-bj-proj 35867 df-bj-1upl 35874 df-bj-2upl 35887 df-bj-pr2 35891 |
This theorem is referenced by: bj-2uplth 35897 bj-2uplex 35898 |
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