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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 5825 | . 2 ⊢ dom (◡𝐵 ∘ ◡𝐴) ⊆ dom ◡𝐴 | |
2 | df-rn 5547 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵) | |
3 | cnvco 5739 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
4 | 3 | dmeqi 5758 | . . 3 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
5 | 2, 4 | eqtri 2759 | . 2 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
6 | df-rn 5547 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | 1, 5, 6 | 3sstr4i 3930 | 1 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3853 ◡ccnv 5535 dom cdm 5536 ran crn 5537 ∘ ccom 5540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 |
This theorem is referenced by: cossxp 6115 fcof 6546 fcoOLD 6548 fco3OLD 6557 fin23lem29 9920 fin23lem30 9921 wunco 10312 imasless 16999 gsumzf1o 19251 znleval 20473 pi1xfrcnvlem 23907 pjss1coi 30198 pj3i 30243 smatrcl 31414 mblfinlem3 35502 mblfinlem4 35503 ismblfin 35504 relexp0a 40942 rntrclfv 40958 stoweidlem27 43186 fourierdlem42 43308 hoicvr 43704 |
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