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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 34925 | . . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
2 | bj-pr1eq 34929 | . . 3 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴) |
4 | bj-pr1val 34931 | . 2 ⊢ pr1 ({∅} × tag 𝐴) = if(∅ = ∅, 𝐴, ∅) | |
5 | eqid 2737 | . . 3 ⊢ ∅ = ∅ | |
6 | 5 | iftruei 4446 | . 2 ⊢ if(∅ = ∅, 𝐴, ∅) = 𝐴 |
7 | 3, 4, 6 | 3eqtri 2769 | 1 ⊢ pr1 ⦅𝐴⦆ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∅c0 4237 ifcif 4439 {csn 4541 × cxp 5549 tag bj-ctag 34901 ⦅bj-c1upl 34924 pr1 bj-cpr1 34927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-bj-sngl 34893 df-bj-tag 34902 df-bj-proj 34918 df-bj-1upl 34925 df-bj-pr1 34928 |
This theorem is referenced by: bj-1uplth 34934 bj-1uplex 34935 bj-pr21val 34940 |
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