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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr1val | Structured version Visualization version GIF version |
Description: Value of the first projection. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-pr1val | ⊢ pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-pr1 34210 | . 2 ⊢ pr1 ({𝐴} × tag 𝐵) = (∅ Proj ({𝐴} × tag 𝐵)) | |
2 | 0ex 5202 | . . 3 ⊢ ∅ ∈ V | |
3 | bj-projval 34205 | . . 3 ⊢ (∅ ∈ V → (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (∅ Proj ({𝐴} × tag 𝐵)) = if(𝐴 = ∅, 𝐵, ∅) |
5 | 1, 4 | eqtri 2841 | 1 ⊢ pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 ifcif 4463 {csn 4557 × cxp 5546 tag bj-ctag 34183 Proj bj-cproj 34199 pr1 bj-cpr1 34209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-bj-sngl 34175 df-bj-tag 34184 df-bj-proj 34200 df-bj-pr1 34210 |
This theorem is referenced by: bj-pr11val 34214 bj-pr21val 34222 |
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