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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 6105 | . 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
2 | df-ima 5568 | . 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
3 | 1, 2 | eqtr4i 2847 | 1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ran crn 5556 ↾ cres 5557 “ cima 5558 ∘ ccom 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: dmco 6107 isf34lem7 9801 isf34lem6 9802 imasless 16813 gsumzf1o 19032 gsumzmhm 19057 gsumzinv 19065 dprdf1o 19154 pf1rcl 20512 ovolficcss 24070 volsup 24157 uniiccdif 24179 uniioombllem3 24186 dyadmbl 24201 itg1climres 24315 cvmlift3lem6 32571 mblfinlem2 34945 volsupnfl 34952 |
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