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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 6100 | . 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
2 | df-ima 5563 | . 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 1533 ran crn 5551 ↾ cres 5552 “ cima 5553 ∘ ccom 5554 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 |
This theorem is referenced by: dmco 6102 isf34lem7 9795 isf34lem6 9796 imasless 16807 gsumzf1o 19026 gsumzmhm 19051 gsumzinv 19059 dprdf1o 19148 pf1rcl 20506 ovolficcss 24064 volsup 24151 uniiccdif 24173 uniioombllem3 24180 dyadmbl 24195 itg1climres 24309 cvmlift3lem6 32566 mblfinlem2 34924 volsupnfl 34931 |
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