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Mirrors > Home > MPE Home > Th. List > imass1 | Structured version Visualization version GIF version |
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.) |
Ref | Expression |
---|---|
imass1 | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssres 5344 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) | |
2 | rnss 5275 | . . 3 ⊢ ((𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶) → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) |
4 | df-ima 5051 | . 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
5 | df-ima 5051 | . 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
6 | 3, 4, 5 | 3sstr4g 3609 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3540 ran crn 5039 ↾ cres 5040 “ cima 5041 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-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-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: vdwnnlem1 15537 dprdres 18250 imasnopn 21303 imasncld 21304 imasncls 21305 utoptop 21848 restutop 21851 ustuqtop3 21857 utopreg 21866 metustbl 22181 imadifxp 28796 esum2d 29482 eulerpartlemmf 29764 brtrclfv2 37038 frege97d 37063 frege109d 37068 frege131d 37075 hess 37094 |
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