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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 35817 | . . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
2 | bj-pr1eq 35821 | . . 3 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴) |
4 | bj-pr1val 35823 | . 2 ⊢ pr1 ({∅} × tag 𝐴) = if(∅ = ∅, 𝐴, ∅) | |
5 | eqid 2733 | . . 3 ⊢ ∅ = ∅ | |
6 | 5 | iftruei 4534 | . 2 ⊢ if(∅ = ∅, 𝐴, ∅) = 𝐴 |
7 | 3, 4, 6 | 3eqtri 2765 | 1 ⊢ pr1 ⦅𝐴⦆ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∅c0 4321 ifcif 4527 {csn 4627 × cxp 5673 tag bj-ctag 35793 ⦅bj-c1upl 35816 pr1 bj-cpr1 35819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-bj-sngl 35785 df-bj-tag 35794 df-bj-proj 35810 df-bj-1upl 35817 df-bj-pr1 35820 |
This theorem is referenced by: bj-1uplth 35826 bj-1uplex 35827 bj-pr21val 35832 |
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