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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 36994 | . . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
2 | bj-pr2eq 36999 | . . . 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 37000 | . . 3 ⊢ pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) | |
5 | 3, 4 | eqtri 2763 | . 2 ⊢ pr2 ⦅𝐴, 𝐵⦆ = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) |
6 | df-bj-1upl 36981 | . . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
7 | bj-pr2eq 36999 | . . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴) |
9 | bj-pr2val 37001 | . . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅) | |
10 | 1n0 8525 | . . . . . 6 ⊢ 1o ≠ ∅ | |
11 | 10 | nesymi 2996 | . . . . 5 ⊢ ¬ ∅ = 1o |
12 | 11 | iffalsei 4541 | . . . 4 ⊢ if(∅ = 1o, 𝐴, ∅) = ∅ |
13 | 8, 9, 12 | 3eqtri 2767 | . . 3 ⊢ pr2 ⦅𝐴⦆ = ∅ |
14 | bj-pr2val 37001 | . . . 4 ⊢ pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅) | |
15 | eqid 2735 | . . . . 5 ⊢ 1o = 1o | |
16 | 15 | iftruei 4538 | . . . 4 ⊢ if(1o = 1o, 𝐵, ∅) = 𝐵 |
17 | 14, 16 | eqtri 2763 | . . 3 ⊢ pr2 ({1o} × tag 𝐵) = 𝐵 |
18 | 13, 17 | uneq12i 4176 | . 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵) |
19 | 0un 4402 | . 2 ⊢ (∅ ∪ 𝐵) = 𝐵 | |
20 | 5, 18, 19 | 3eqtri 2767 | 1 ⊢ pr2 ⦅𝐴, 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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