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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 5832 | . . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) | |
2 | 1 | rneqi 5771 | . . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) |
3 | rnun 5971 | . . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) | |
4 | 2, 3 | eqtri 2821 | . 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
5 | df-ima 5532 | . 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶)) | |
6 | df-ima 5532 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
7 | df-ima 5532 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
8 | 6, 7 | uneq12i 4088 | . 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
9 | 4, 5, 8 | 3eqtr4i 2831 | 1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∪ cun 3879 ran crn 5520 ↾ cres 5521 “ cima 5522 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: fnimapr 6722 domunfican 8775 fiint 8779 fodomfi 8781 marypha1lem 8881 resunimafz0 13799 dprd2da 19157 dmdprdsplit2lem 19160 uniioombllem3 24189 mbfimaicc 24235 plyeq0 24808 fnimatp 30440 ffsrn 30491 tocyccntz 30836 noetalem4 33333 imadifss 35032 poimirlem1 35058 poimirlem2 35059 poimirlem3 35060 poimirlem4 35061 poimirlem6 35063 poimirlem7 35064 poimirlem11 35068 poimirlem12 35069 poimirlem15 35072 poimirlem16 35073 poimirlem17 35074 poimirlem19 35076 poimirlem20 35077 poimirlem23 35080 poimirlem24 35081 poimirlem25 35082 poimirlem29 35086 poimirlem31 35088 mbfposadd 35104 itg2addnclem2 35109 ftc1anclem1 35130 ftc1anclem5 35134 brtrclfv2 40428 frege77d 40447 frege109d 40458 frege131d 40465 dffrege76 40640 icccncfext 42529 |
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