Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pr1un | Structured version Visualization version GIF version |
Description: The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-pr1un | ⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-projun 34203 | . 2 ⊢ (∅ Proj (𝐴 ∪ 𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵)) | |
2 | df-bj-pr1 34210 | . 2 ⊢ pr1 (𝐴 ∪ 𝐵) = (∅ Proj (𝐴 ∪ 𝐵)) | |
3 | df-bj-pr1 34210 | . . 3 ⊢ pr1 𝐴 = (∅ Proj 𝐴) | |
4 | df-bj-pr1 34210 | . . 3 ⊢ pr1 𝐵 = (∅ Proj 𝐵) | |
5 | 3, 4 | uneq12i 4134 | . 2 ⊢ (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵)) |
6 | 1, 2, 5 | 3eqtr4i 2851 | 1 ⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∪ cun 3931 ∅c0 4288 Proj bj-cproj 34199 pr1 bj-cpr1 34209 |
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-pr1 34210 |
This theorem is referenced by: bj-pr21val 34222 |
Copyright terms: Public domain | W3C validator |