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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 36604 | . 2 ⊢ (∅ Proj (𝐴 ∪ 𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵)) | |
2 | df-bj-pr1 36611 | . 2 ⊢ pr1 (𝐴 ∪ 𝐵) = (∅ Proj (𝐴 ∪ 𝐵)) | |
3 | df-bj-pr1 36611 | . . 3 ⊢ pr1 𝐴 = (∅ Proj 𝐴) | |
4 | df-bj-pr1 36611 | . . 3 ⊢ pr1 𝐵 = (∅ Proj 𝐵) | |
5 | 3, 4 | uneq12i 4158 | . 2 ⊢ (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵)) |
6 | 1, 2, 5 | 3eqtr4i 2763 | 1 ⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3942 ∅c0 4322 Proj bj-cproj 36600 pr1 bj-cpr1 36610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-bj-proj 36601 df-bj-pr1 36611 |
This theorem is referenced by: bj-pr21val 36623 |
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