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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 5962 | . 2 ⊢ dom (◡𝐵 ∘ ◡𝐴) ⊆ dom ◡𝐴 | |
2 | df-rn 5680 | . . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵) | |
3 | cnvco 5877 | . . . 4 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴) | |
4 | 3 | dmeqi 5896 | . . 3 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
5 | 2, 4 | eqtri 2759 | . 2 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴) |
6 | df-rn 5680 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | 1, 5, 6 | 3sstr4i 4021 | 1 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3944 ◡ccnv 5668 dom cdm 5669 ran crn 5670 ∘ ccom 5673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 |
This theorem is referenced by: cossxp 6260 fcof 6727 fcoOLD 6729 fco3OLD 6738 fin23lem29 10318 fin23lem30 10319 wunco 10710 imasless 17468 gsumzf1o 19739 znleval 21043 pi1xfrcnvlem 24501 pjss1coi 31279 pj3i 31324 smatrcl 32607 mblfinlem3 36331 mblfinlem4 36332 ismblfin 36333 relexp0a 42238 rntrclfv 42254 stoweidlem27 44516 fourierdlem42 44638 hoicvr 45037 |
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