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Mirrors > Home > MPE Home > Th. List > imaundi | Structured version Visualization version GIF version |
Description: Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
imaundi | ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resundi 5994 | . . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) | |
2 | 1 | rneqi 5935 | . . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) |
3 | rnun 6144 | . . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) | |
4 | 2, 3 | eqtri 2758 | . 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
5 | df-ima 5688 | . 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶)) | |
6 | df-ima 5688 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
7 | df-ima 5688 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
8 | 6, 7 | uneq12i 4160 | . 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
9 | 4, 5, 8 | 3eqtr4i 2768 | 1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∪ cun 3945 ran crn 5676 ↾ cres 5677 “ cima 5678 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 |
This theorem is referenced by: cnvimassrndm 6150 fnimapr 6974 naddasslem1 8695 naddasslem2 8696 imafi 9177 domunfican 9322 fiint 9326 fodomfi 9327 marypha1lem 9430 resunimafz0 14408 dprd2da 19953 dmdprdsplit2lem 19956 uniioombllem3 25334 mbfimaicc 25380 plyeq0 25960 madeoldsuc 27616 negsbdaylem 27769 fnimatp 32169 ffsrn 32221 tocyccntz 32573 imadifss 36766 poimirlem1 36792 poimirlem2 36793 poimirlem3 36794 poimirlem4 36795 poimirlem6 36797 poimirlem7 36798 poimirlem11 36802 poimirlem12 36803 poimirlem15 36806 poimirlem16 36807 poimirlem17 36808 poimirlem19 36810 poimirlem20 36811 poimirlem23 36814 poimirlem24 36815 poimirlem25 36816 poimirlem29 36820 poimirlem31 36822 mbfposadd 36838 itg2addnclem2 36843 ftc1anclem1 36864 ftc1anclem5 36868 brtrclfv2 42780 frege77d 42799 frege109d 42810 frege131d 42817 dffrege76 42992 icccncfext 44901 |
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