![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 35801 | . 2 ⊢ pr2 ({𝐴} × tag 𝐵) = (1o Proj ({𝐴} × tag 𝐵)) | |
2 | 1oex 8463 | . . 3 ⊢ 1o ∈ V | |
3 | bj-projval 35782 | . . 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 2761 | 1 ⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1o, 𝐵, ∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4320 ifcif 4524 {csn 4624 × cxp 5670 1oc1o 8446 tag bj-ctag 35760 Proj bj-cproj 35776 pr2 bj-cpr2 35800 |
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 5295 ax-nul 5302 ax-pr 5423 |
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 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-suc 6362 df-1o 8453 df-bj-sngl 35752 df-bj-tag 35761 df-bj-proj 35777 df-bj-pr2 35801 |
This theorem is referenced by: bj-pr22val 35805 |
Copyright terms: Public domain | W3C validator |