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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 34807 | . 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) | |
2 | df-bj-pr2 34828 | . 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵)) | |
3 | df-bj-pr2 34828 | . . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
4 | df-bj-pr2 34828 | . . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵) | |
5 | 3, 4 | uneq12i 4051 | . 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) |
6 | 1, 2, 5 | 3eqtr4i 2771 | 1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cun 3841 1oc1o 8124 Proj bj-cproj 34803 pr2 bj-cpr2 34827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-rab 3062 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-cnv 5533 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-bj-proj 34804 df-bj-pr2 34828 |
This theorem is referenced by: bj-pr22val 34832 |
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