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Mirrors > Home > MPE Home > Th. List > imass2 | Structured version Visualization version GIF version |
Description: Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
imass2 | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssres2 5635 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) | |
2 | rnss 5557 | . . 3 ⊢ ((𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵) → ran (𝐶 ↾ 𝐴) ⊆ ran (𝐶 ↾ 𝐵)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐶 ↾ 𝐴) ⊆ ran (𝐶 ↾ 𝐵)) |
4 | df-ima 5325 | . 2 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴) | |
5 | df-ima 5325 | . 2 ⊢ (𝐶 “ 𝐵) = ran (𝐶 ↾ 𝐵) | |
6 | 3, 4, 5 | 3sstr4g 3842 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3769 ran crn 5313 ↾ cres 5314 “ cima 5315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-cnv 5320 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 |
This theorem is referenced by: funimass1 6182 funimass2 6183 fvimacnv 6558 f1imass 6749 ecinxp 8060 sbthlem1 8312 sbthlem2 8313 php3 8388 ordtypelem2 8666 tcrank 8997 limsupgord 14544 isercoll 14739 isacs1i 16632 gsumzf1o 18628 dprdres 18743 dprd2da 18757 dmdprdsplit2lem 18760 lmhmlsp 19370 f1lindf 20486 iscnp4 21396 cnpco 21400 cncls2i 21403 cnntri 21404 cnrest2 21419 cnpresti 21421 cnprest 21422 1stcfb 21577 xkococnlem 21791 qtopval2 21828 tgqtop 21844 qtoprest 21849 kqdisj 21864 regr1lem 21871 kqreglem1 21873 kqreglem2 21874 kqnrmlem1 21875 kqnrmlem2 21876 nrmhmph 21926 fbasrn 22016 elfm2 22080 fmfnfmlem1 22086 fmco 22093 flffbas 22127 cnpflf2 22132 cnextcn 22199 metcnp3 22673 metustto 22686 cfilucfil 22692 uniioombllem3 23693 dyadmbllem 23707 mbfconstlem 23735 i1fima2 23787 itg2gt0 23868 ellimc3 23984 limcflf 23986 limcresi 23990 limciun 23999 lhop 24120 ig1peu 24272 ig1pdvds 24277 psercnlem2 24519 dvloglem 24735 efopn 24745 txomap 30417 tpr2rico 30474 cvmsss2 31773 cvmopnlem 31777 cvmliftmolem1 31780 cvmliftlem15 31797 cvmlift2lem9 31810 imadifss 33873 poimirlem1 33899 poimirlem2 33900 poimirlem3 33901 poimirlem15 33913 poimirlem30 33928 dvtan 33948 heibor1lem 34095 isnumbasabl 38457 isnumbasgrp 38458 dfacbasgrp 38459 trclimalb2 38797 frege81d 38818 fnfvimad 40203 fnfvima2 40221 imass2d 40223 limccog 40592 liminfgord 40726 |
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