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Mathbox for Glauco Siliprandi |
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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 6014 | . . 3 ⊢ (𝐹 ↾ (𝐴 ∪ 𝐵)) = ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵)) | |
2 | 1 | rneqi 5951 | . 2 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵)) |
3 | rnun 6168 | . 2 ⊢ ran ((𝐹 ↾ 𝐴) ∪ (𝐹 ↾ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) | |
4 | 2, 3 | eqtri 2763 | 1 ⊢ ran (𝐹 ↾ (𝐴 ∪ 𝐵)) = (ran (𝐹 ↾ 𝐴) ∪ ran (𝐹 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3961 ran crn 5690 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 |
This theorem is referenced by: sge0split 46365 |
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