Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5870 | . . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) | |
2 | 1 | rneqi 5810 | . . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) |
3 | rnun 6007 | . . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) | |
4 | 2, 3 | eqtri 2847 | . 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
5 | df-ima 5571 | . 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶)) | |
6 | df-ima 5571 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
7 | df-ima 5571 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
8 | 6, 7 | uneq12i 4140 | . 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶)) |
9 | 4, 5, 8 | 3eqtr4i 2857 | 1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∪ cun 3937 ran crn 5559 ↾ cres 5560 “ cima 5561 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 |
This theorem is referenced by: fnimapr 6750 domunfican 8794 fiint 8798 fodomfi 8800 marypha1lem 8900 resunimafz0 13806 dprd2da 19167 dmdprdsplit2lem 19170 uniioombllem3 24189 mbfimaicc 24235 plyeq0 24804 fnimatp 30426 ffsrn 30468 tocyccntz 30790 noetalem4 33224 imadifss 34871 poimirlem1 34897 poimirlem2 34898 poimirlem3 34899 poimirlem4 34900 poimirlem6 34902 poimirlem7 34903 poimirlem11 34907 poimirlem12 34908 poimirlem15 34911 poimirlem16 34912 poimirlem17 34913 poimirlem19 34915 poimirlem20 34916 poimirlem23 34919 poimirlem24 34920 poimirlem25 34921 poimirlem29 34925 poimirlem31 34927 mbfposadd 34943 itg2addnclem2 34948 ftc1anclem1 34971 ftc1anclem5 34975 brtrclfv2 40078 frege77d 40097 frege109d 40108 frege131d 40115 dffrege76 40291 icccncfext 42176 |
Copyright terms: Public domain | W3C validator |