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Mirrors > Home > MPE Home > Th. List > rnco2 | Structured version Visualization version GIF version |
Description: The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
Ref | Expression |
---|---|
rnco2 | ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnco 6274 | . 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
2 | df-ima 5702 | . 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
3 | 1, 2 | eqtr4i 2766 | 1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ran crn 5690 ↾ cres 5691 “ cima 5692 ∘ ccom 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: dmco 6276 isf34lem7 10417 isf34lem6 10418 imasless 17587 gsumzf1o 19945 gsumzmhm 19970 gsumzinv 19978 dprdf1o 20067 pf1rcl 22369 ovolficcss 25518 volsup 25605 uniiccdif 25627 uniioombllem3 25634 dyadmbl 25649 itg1climres 25764 cvmlift3lem6 35309 mblfinlem2 37645 volsupnfl 37652 |
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