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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr2un | Structured version Visualization version GIF version |
Description: The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-pr2un | ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-projun 34203 | . 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) | |
2 | df-bj-pr2 34224 | . 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵)) | |
3 | df-bj-pr2 34224 | . . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
4 | df-bj-pr2 34224 | . . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵) | |
5 | 3, 4 | uneq12i 4134 | . 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) |
6 | 1, 2, 5 | 3eqtr4i 2851 | 1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∪ cun 3931 1oc1o 8084 Proj bj-cproj 34199 pr2 bj-cpr2 34223 |
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 |
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-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 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-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-bj-proj 34200 df-bj-pr2 34224 |
This theorem is referenced by: bj-pr22val 34228 |
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