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| Mirrors > Home > MPE Home > Th. List > rnuni | Structured version Visualization version GIF version | ||
| Description: The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.) |
| Ref | Expression |
|---|---|
| rnuni | ⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uniiun 5032 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
| 2 | 1 | rneqi 5915 | . 2 ⊢ ran ∪ 𝐴 = ran ∪ 𝑥 ∈ 𝐴 𝑥 |
| 3 | rniun 6134 | . 2 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝑥 = ∪ 𝑥 ∈ 𝐴 ran 𝑥 | |
| 4 | 2, 3 | eqtri 2757 | 1 ⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1539 ∪ cuni 4881 ∪ ciun 4965 ran crn 5653 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-ext 2706 ax-sep 5264 ax-nul 5274 ax-pr 5400 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2713 df-cleq 2726 df-clel 2808 df-rex 3060 df-rab 3414 df-v 3459 df-dif 3927 df-un 3929 df-ss 3941 df-nul 4307 df-if 4499 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4882 df-iun 4967 df-br 5118 df-opab 5180 df-cnv 5660 df-dm 5662 df-rn 5663 |
| This theorem is referenced by: ackbij2 10249 axdc3lem2 10458 unirnmap 45166 unirnmapsn 45172 |
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