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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr2val | Structured version Visualization version GIF version |
Description: Value of the second projection. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-pr2val | ⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-pr2 35891 | . 2 ⊢ pr2 ({𝐴} × tag 𝐵) = (1o Proj ({𝐴} × tag 𝐵)) | |
2 | 1oex 8475 | . . 3 ⊢ 1o ∈ V | |
3 | bj-projval 35872 | . . 3 ⊢ (1o ∈ V → (1o Proj ({𝐴} × tag 𝐵)) = if(𝐴 = 1o, 𝐵, ∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (1o Proj ({𝐴} × tag 𝐵)) = if(𝐴 = 1o, 𝐵, ∅) |
5 | 1, 4 | eqtri 2760 | 1 ⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 ifcif 4528 {csn 4628 × cxp 5674 1oc1o 8458 tag bj-ctag 35850 Proj bj-cproj 35866 pr2 bj-cpr2 35890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-suc 6370 df-1o 8465 df-bj-sngl 35842 df-bj-tag 35851 df-bj-proj 35867 df-bj-pr2 35891 |
This theorem is referenced by: bj-pr22val 35895 |
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