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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 33932 | . 2 ⊢ (∅ Proj (𝐴 ∪ 𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵)) | |
2 | df-bj-pr1 33939 | . 2 ⊢ pr1 (𝐴 ∪ 𝐵) = (∅ Proj (𝐴 ∪ 𝐵)) | |
3 | df-bj-pr1 33939 | . . 3 ⊢ pr1 𝐴 = (∅ Proj 𝐴) | |
4 | df-bj-pr1 33939 | . . 3 ⊢ pr1 𝐵 = (∅ Proj 𝐵) | |
5 | 3, 4 | uneq12i 4064 | . 2 ⊢ (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵)) |
6 | 1, 2, 5 | 3eqtr4i 2831 | 1 ⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1525 ∪ cun 3863 ∅c0 4217 Proj bj-cproj 33928 pr1 bj-cpr1 33938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-br 4969 df-opab 5031 df-cnv 5458 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-bj-proj 33929 df-bj-pr1 33939 |
This theorem is referenced by: bj-pr21val 33951 |
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