|   | Mathbox for BJ | < Previous  
      Next > Nearby theorems | |
| 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 36995 | . 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) | |
| 2 | df-bj-pr2 37016 | . 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵)) | |
| 3 | df-bj-pr2 37016 | . . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
| 4 | df-bj-pr2 37016 | . . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵) | |
| 5 | 3, 4 | uneq12i 4166 | . 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) | 
| 6 | 1, 2, 5 | 3eqtr4i 2775 | 1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∪ cun 3949 1oc1o 8499 Proj bj-cproj 36991 pr2 bj-cpr2 37015 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-bj-proj 36992 df-bj-pr2 37016 | 
| This theorem is referenced by: bj-pr22val 37020 | 
| Copyright terms: Public domain | W3C validator |