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Mirrors > Home > MPE Home > Th. List > rncoss | Structured version Visualization version GIF version |
Description: Range of a composition. (Contributed by NM, 19-Mar-1998.) |
Ref | Expression |
---|---|
rncoss | ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmcoss 5844 | . 2 ⊢ dom (◡𝐵 ∘ ◡𝐴) ⊆ dom ◡𝐴 | |
2 | df-rn 5568 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵) | |
3 | cnvco 5758 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
4 | 3 | dmeqi 5775 | . . 3 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
5 | 2, 4 | eqtri 2846 | . 2 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
6 | df-rn 5568 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | 1, 5, 6 | 3sstr4i 4012 | 1 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3938 ◡ccnv 5556 dom cdm 5557 ran crn 5558 ∘ ccom 5561 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 |
This theorem is referenced by: cossxp 6125 fco 6533 fin23lem29 9765 fin23lem30 9766 wunco 10157 imasless 16815 gsumzf1o 19034 znleval 20703 pi1xfrcnvlem 23662 pjss1coi 29942 pj3i 29987 smatrcl 31063 mblfinlem3 34933 mblfinlem4 34934 ismblfin 34935 relexp0a 40068 rntrclfv 40084 fco3 41498 stoweidlem27 42319 fourierdlem42 42441 hoicvr 42837 |
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