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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 36551 | . 2 ⊢ pr2 ({𝐴} × tag 𝐵) = (1o Proj ({𝐴} × tag 𝐵)) | |
2 | 1oex 8495 | . . 3 ⊢ 1o ∈ V | |
3 | bj-projval 36532 | . . 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 2753 | 1 ⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∅c0 4318 ifcif 4524 {csn 4624 × cxp 5670 1oc1o 8478 tag bj-ctag 36510 Proj bj-cproj 36526 pr2 bj-cpr2 36550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-suc 6370 df-1o 8485 df-bj-sngl 36502 df-bj-tag 36511 df-bj-proj 36527 df-bj-pr2 36551 |
This theorem is referenced by: bj-pr22val 36555 |
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