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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr11val | Structured version Visualization version GIF version | ||
| Description: Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| bj-pr11val | ⊢ pr1 ⦅𝐴⦆ = 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-bj-1upl 37000 | . . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
| 2 | bj-pr1eq 37004 | . . 3 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴)) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴) | 
| 4 | bj-pr1val 37006 | . 2 ⊢ pr1 ({∅} × tag 𝐴) = if(∅ = ∅, 𝐴, ∅) | |
| 5 | eqid 2736 | . . 3 ⊢ ∅ = ∅ | |
| 6 | 5 | iftruei 4531 | . 2 ⊢ if(∅ = ∅, 𝐴, ∅) = 𝐴 | 
| 7 | 3, 4, 6 | 3eqtri 2768 | 1 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∅c0 4332 ifcif 4524 {csn 4625 × cxp 5682 tag bj-ctag 36976 ⦅bj-c1upl 36999 pr1 bj-cpr1 37002 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-bj-sngl 36968 df-bj-tag 36977 df-bj-proj 36993 df-bj-1upl 37000 df-bj-pr1 37003 | 
| This theorem is referenced by: bj-1uplth 37009 bj-1uplex 37010 bj-pr21val 37015 | 
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