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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 5385 | . 2 ⊢ dom (◡𝐵 ∘ ◡𝐴) ⊆ dom ◡𝐴 | |
2 | df-rn 5125 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵) | |
3 | cnvco 5308 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
4 | 3 | dmeqi 5325 | . . 3 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
5 | 2, 4 | eqtri 2644 | . 2 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
6 | df-rn 5125 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | 1, 5, 6 | 3sstr4i 3644 | 1 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3574 ◡ccnv 5113 dom cdm 5114 ran crn 5115 ∘ ccom 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 |
This theorem is referenced by: cossxp 5658 fco 6058 fin23lem29 9163 fin23lem30 9164 wunco 9555 imasless 16200 gsumzf1o 18313 znleval 19903 pi1xfrcnvlem 22856 pjss1coi 29022 pj3i 29067 smatrcl 29862 mblfinlem3 33448 mblfinlem4 33449 ismblfin 33450 relexp0a 38008 rntrclfv 38024 fco3 39421 stoweidlem27 40244 fourierdlem42 40366 hoicvr 40762 |
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