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 32196 | . 2 ⊢ pr2 ({𝐴} × tag 𝐵) = (1𝑜 Proj ({𝐴} × tag 𝐵)) | |
2 | bj-1ex 32131 | . . 3 ⊢ 1𝑜 ∈ V | |
3 | bj-projval 32177 | . . 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 2632 | 1 ⊢ pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1𝑜, 𝐵, ∅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ifcif 4036 {csn 4125 × cxp 5036 1𝑜c1o 7440 tag bj-ctag 32155 Proj bj-cproj 32171 pr2 bj-cpr2 32195 |
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-8 1979 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 ax-un 6847 |
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-uni 4373 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-suc 5646 df-1o 7447 df-bj-sngl 32147 df-bj-tag 32156 df-bj-proj 32172 df-bj-pr2 32196 |
This theorem is referenced by: bj-pr22val 32200 |
Copyright terms: Public domain | W3C validator |