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| Mirrors > Home > MPE Home > Th. List > imaundir | Structured version Visualization version GIF version | ||
| Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.) |
| Ref | Expression |
|---|---|
| imaundir | ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ima 5698 | . . 3 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ran ((𝐴 ∪ 𝐵) ↾ 𝐶) | |
| 2 | resundir 6012 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) | |
| 3 | 2 | rneqi 5948 | . . 3 ⊢ ran ((𝐴 ∪ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) |
| 4 | rnun 6165 | . . 3 ⊢ ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶)) | |
| 5 | 1, 3, 4 | 3eqtri 2769 | . 2 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶)) |
| 6 | df-ima 5698 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
| 7 | df-ima 5698 | . . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
| 8 | 6, 7 | uneq12i 4166 | . 2 ⊢ ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶)) |
| 9 | 5, 8 | eqtr4i 2768 | 1 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cun 3949 ran crn 5686 ↾ cres 5687 “ cima 5688 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: fvun 6999 suppun 8209 fsuppun 9427 fpwwe2lem12 10682 ustuqtop1 24250 mbfres2 25680 imadifxp 32614 suppun2 32693 eulerpartlemt 34373 bj-projun 36995 bj-funun 37253 poimirlem3 37630 poimirlem15 37642 brtrclfv2 43740 frege131d 43777 unhe1 43798 frege110 43986 frege133 44009 aacllem 49320 |
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