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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 5024 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
| 2 | 1 | rneqi 5903 | . 2 ⊢ ran ∪ 𝐴 = ran ∪ 𝑥 ∈ 𝐴 𝑥 |
| 3 | rniun 6122 | . 2 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝑥 = ∪ 𝑥 ∈ 𝐴 ran 𝑥 | |
| 4 | 2, 3 | eqtri 2753 | 1 ⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cuni 4873 ∪ ciun 4957 ran crn 5641 |
| 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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-11 2158 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pr 5389 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-ss 3933 df-nul 4299 df-if 4491 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-cnv 5648 df-dm 5650 df-rn 5651 |
| This theorem is referenced by: ackbij2 10201 axdc3lem2 10410 unirnmap 45195 unirnmapsn 45201 |
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