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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 34310 | . 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) | |
2 | df-bj-pr2 34331 | . 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵)) | |
3 | df-bj-pr2 34331 | . . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴) | |
4 | df-bj-pr2 34331 | . . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵) | |
5 | 3, 4 | uneq12i 4140 | . 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵)) |
6 | 1, 2, 5 | 3eqtr4i 2857 | 1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∪ cun 3937 1oc1o 8098 Proj bj-cproj 34306 pr2 bj-cpr2 34330 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-bj-proj 34307 df-bj-pr2 34331 |
This theorem is referenced by: bj-pr22val 34335 |
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