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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 6156 | . 2 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
2 | df-ima 5602 | . 2 ⊢ (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵) | |
3 | 1, 2 | eqtr4i 2769 | 1 ⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ran crn 5590 ↾ cres 5591 “ cima 5592 ∘ ccom 5593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 |
This theorem is referenced by: dmco 6158 isf34lem7 10135 isf34lem6 10136 imasless 17251 gsumzf1o 19513 gsumzmhm 19538 gsumzinv 19546 dprdf1o 19635 pf1rcl 21515 ovolficcss 24633 volsup 24720 uniiccdif 24742 uniioombllem3 24749 dyadmbl 24764 itg1climres 24879 cvmlift3lem6 33286 mblfinlem2 35815 volsupnfl 35822 |
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