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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 4945 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
2 | 1 | rneqi 5771 | . 2 ⊢ ran ∪ 𝐴 = ran ∪ 𝑥 ∈ 𝐴 𝑥 |
3 | rniun 5973 | . 2 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝑥 = ∪ 𝑥 ∈ 𝐴 ran 𝑥 | |
4 | 2, 3 | eqtri 2821 | 1 ⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∪ cuni 4800 ∪ ciun 4881 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: ackbij2 9654 axdc3lem2 9862 unirnmap 41837 unirnmapsn 41843 |
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