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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 32179 | . . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
2 | bj-pr1eq 32183 | . . 3 ⊢ (⦅𝐴⦆ = ({∅} × tag 𝐴) → pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ pr1 ⦅𝐴⦆ = pr1 ({∅} × tag 𝐴) |
4 | bj-pr1val 32185 | . 2 ⊢ pr1 ({∅} × tag 𝐴) = if(∅ = ∅, 𝐴, ∅) | |
5 | eqid 2610 | . . 3 ⊢ ∅ = ∅ | |
6 | 5 | iftruei 4043 | . 2 ⊢ if(∅ = ∅, 𝐴, ∅) = 𝐴 |
7 | 3, 4, 6 | 3eqtri 2636 | 1 ⊢ pr1 ⦅𝐴⦆ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∅c0 3874 ifcif 4036 {csn 4125 × cxp 5036 tag bj-ctag 32155 ⦅bj-c1upl 32178 pr1 bj-cpr1 32181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-bj-sngl 32147 df-bj-tag 32156 df-bj-proj 32172 df-bj-1upl 32179 df-bj-pr1 32182 |
This theorem is referenced by: bj-1uplth 32188 bj-1uplex 32189 bj-pr21val 32194 |
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