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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 5644 | . . 3 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ran ((𝐴 ∪ 𝐵) ↾ 𝐶) | |
2 | resundir 5950 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) | |
3 | 2 | rneqi 5890 | . . 3 ⊢ ran ((𝐴 ∪ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) |
4 | rnun 6096 | . . 3 ⊢ ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶)) | |
5 | 1, 3, 4 | 3eqtri 2768 | . 2 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶)) |
6 | df-ima 5644 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
7 | df-ima 5644 | . . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
8 | 6, 7 | uneq12i 4119 | . 2 ⊢ ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶)) |
9 | 5, 8 | eqtr4i 2767 | 1 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∪ cun 3906 ran crn 5632 ↾ cres 5633 “ cima 5634 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-cnv 5639 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 |
This theorem is referenced by: fvun 6928 suppun 8111 fsuppun 9320 fpwwe2lem12 10574 ustuqtop1 23577 mbfres2 24993 imadifxp 31405 eulerpartlemt 32840 bj-projun 35432 bj-funun 35690 poimirlem3 36048 poimirlem15 36060 brtrclfv2 41941 frege131d 41978 unhe1 41999 frege110 42187 frege133 42210 aacllem 47180 |
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