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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 5330 | . . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) | |
2 | 1 | rneqi 5273 | . . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) |
3 | rnun 5460 | . . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) | |
4 | 2, 3 | eqtri 2632 | . 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
5 | df-ima 5051 | . 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶)) | |
6 | df-ima 5051 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
7 | df-ima 5051 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
8 | 6, 7 | uneq12i 3727 | . 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
9 | 4, 5, 8 | 3eqtr4i 2642 | 1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 ran crn 5039 ↾ cres 5040 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: fnimapr 6172 domunfican 8118 fiint 8122 fodomfi 8124 marypha1lem 8222 dprd2da 18264 dmdprdsplit2lem 18267 uniioombllem3 23159 mbfimaicc 23206 plyeq0 23771 eupath2lem3 26506 ffsrn 28892 imadifss 32554 poimirlem1 32580 poimirlem2 32581 poimirlem3 32582 poimirlem4 32583 poimirlem6 32585 poimirlem7 32586 poimirlem11 32590 poimirlem12 32591 poimirlem15 32594 poimirlem16 32595 poimirlem17 32596 poimirlem19 32598 poimirlem20 32599 poimirlem23 32602 poimirlem24 32603 poimirlem25 32604 poimirlem29 32608 poimirlem31 32610 mbfposadd 32627 itg2addnclem2 32632 ftc1anclem1 32655 ftc1anclem5 32659 brtrclfv2 37038 frege77d 37057 frege109d 37068 frege131d 37075 dffrege76 37253 icccncfext 38773 resunimafz0 40368 |
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