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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 36996 | . 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) | |
| 2 | df-bj-pr2 37017 | . 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵)) | |
| 3 | df-bj-pr2 37017 | . . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
| 4 | df-bj-pr2 37017 | . . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵) | |
| 5 | 3, 4 | uneq12i 4165 | . 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) | 
| 6 | 1, 2, 5 | 3eqtr4i 2774 | 1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∪ cun 3948 1oc1o 8500 Proj bj-cproj 36992 pr2 bj-cpr2 37016 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-bj-proj 36993 df-bj-pr2 37017 | 
| This theorem is referenced by: bj-pr22val 37021 | 
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