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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 6011 | . . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) | |
| 2 | 1 | rneqi 5948 | . . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) |
| 3 | rnun 6165 | . . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) | |
| 4 | 2, 3 | eqtri 2765 | . 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
| 5 | df-ima 5698 | . 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶)) | |
| 6 | df-ima 5698 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
| 7 | df-ima 5698 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
| 8 | 6, 7 | uneq12i 4166 | . 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
| 9 | 4, 5, 8 | 3eqtr4i 2775 | 1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 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-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: cnvimassrndm 6172 fnimapr 6992 fnimatpd 6993 naddasslem1 8732 naddasslem2 8733 fodomfi 9350 imafiOLD 9354 domunfican 9361 fiint 9366 fiintOLD 9367 fodomfiOLD 9370 marypha1lem 9473 resunimafz0 14484 dprd2da 20062 dmdprdsplit2lem 20065 uniioombllem3 25620 mbfimaicc 25666 plyeq0 26250 madeoldsuc 27923 addsbday 28050 negsbdaylem 28088 pw2bday 28418 zs12bday 28424 ffsrn 32740 tocyccntz 33164 imadifss 37602 poimirlem1 37628 poimirlem2 37629 poimirlem3 37630 poimirlem4 37631 poimirlem6 37633 poimirlem7 37634 poimirlem11 37638 poimirlem12 37639 poimirlem15 37642 poimirlem16 37643 poimirlem17 37644 poimirlem19 37646 poimirlem20 37647 poimirlem23 37650 poimirlem24 37651 poimirlem25 37652 poimirlem29 37656 poimirlem31 37658 mbfposadd 37674 itg2addnclem2 37679 ftc1anclem1 37700 ftc1anclem5 37704 brtrclfv2 43740 frege77d 43759 frege109d 43770 frege131d 43777 dffrege76 43952 icccncfext 45902 cycl3grtri 47914 |
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