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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnresun | Structured version Visualization version GIF version |
Description: Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
rnresun | ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resundi 5850 | . . 3 ⊢ (𝐹 ↾ (𝐴 ∪ 𝐵)) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵)) | |
2 | 1 | rneqi 5791 | . 2 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵)) |
3 | rnun 5989 | . 2 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) | |
4 | 2, 3 | eqtri 2759 | 1 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∪ cun 3851 ran crn 5537 ↾ cres 5538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 |
This theorem is referenced by: sge0split 43565 |
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