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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 37012 | . . . 4 ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) | |
| 2 | bj-pr2eq 37017 | . . . 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 37018 | . . 3 ⊢ pr2 (⦅𝐴⦆ ∪ ({1o} × tag 𝐵)) = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) | |
| 5 | 3, 4 | eqtri 2765 | . 2 ⊢ pr2 ⦅𝐴, 𝐵⦆ = (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) | 
| 6 | df-bj-1upl 36999 | . . . . 5 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
| 7 | bj-pr2eq 37017 | . . . . 5 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴)) | |
| 8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ pr2 ⦅𝐴⦆ = pr2 ({∅} × tag 𝐴) | 
| 9 | bj-pr2val 37019 | . . . 4 ⊢ pr2 ({∅} × tag 𝐴) = if(∅ = 1o, 𝐴, ∅) | |
| 10 | 1n0 8526 | . . . . . 6 ⊢ 1o ≠ ∅ | |
| 11 | 10 | nesymi 2998 | . . . . 5 ⊢ ¬ ∅ = 1o | 
| 12 | 11 | iffalsei 4535 | . . . 4 ⊢ if(∅ = 1o, 𝐴, ∅) = ∅ | 
| 13 | 8, 9, 12 | 3eqtri 2769 | . . 3 ⊢ pr2 ⦅𝐴⦆ = ∅ | 
| 14 | bj-pr2val 37019 | . . . 4 ⊢ pr2 ({1o} × tag 𝐵) = if(1o = 1o, 𝐵, ∅) | |
| 15 | eqid 2737 | . . . . 5 ⊢ 1o = 1o | |
| 16 | 15 | iftruei 4532 | . . . 4 ⊢ if(1o = 1o, 𝐵, ∅) = 𝐵 | 
| 17 | 14, 16 | eqtri 2765 | . . 3 ⊢ pr2 ({1o} × tag 𝐵) = 𝐵 | 
| 18 | 13, 17 | uneq12i 4166 | . 2 ⊢ (pr2 ⦅𝐴⦆ ∪ pr2 ({1o} × tag 𝐵)) = (∅ ∪ 𝐵) | 
| 19 | 0un 4396 | . 2 ⊢ (∅ ∪ 𝐵) = 𝐵 | |
| 20 | 5, 18, 19 | 3eqtri 2769 | 1 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∪ cun 3949 ∅c0 4333 ifcif 4525 {csn 4626 × cxp 5683 1oc1o 8499 tag bj-ctag 36975 ⦅bj-c1upl 36998 ⦅bj-c2uple 37011 pr2 bj-cpr2 37015 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-1o 8506 df-bj-sngl 36967 df-bj-tag 36976 df-bj-proj 36992 df-bj-1upl 36999 df-bj-2upl 37012 df-bj-pr2 37016 | 
| This theorem is referenced by: bj-2uplth 37022 bj-2uplex 37023 | 
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