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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 5306 | . 2 ⊢ dom (◡𝐵 ∘ ◡𝐴) ⊆ dom ◡𝐴 | |
2 | df-rn 5049 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵) | |
3 | cnvco 5230 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
4 | 3 | dmeqi 5247 | . . 3 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
5 | 2, 4 | eqtri 2632 | . 2 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
6 | df-rn 5049 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | 1, 5, 6 | 3sstr4i 3607 | 1 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3540 ◡ccnv 5037 dom cdm 5038 ran crn 5039 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 |
This theorem is referenced by: cossxp 5575 fco 5971 fin23lem29 9046 fin23lem30 9047 wunco 9434 imasless 16023 gsumzf1o 18136 znleval 19722 pi1xfrcnvlem 22664 pjss1coi 28406 pj3i 28451 smatrcl 29190 mblfinlem3 32618 mblfinlem4 32619 ismblfin 32620 relexp0a 37027 rntrclfv 37043 fco3 38416 stoweidlem27 38920 fourierdlem42 39042 hoicvr 39438 |
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