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Mirrors > Home > ILE Home > Th. List > imass2 | 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 4936 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) | |
2 | rnss 4859 | . . 3 ⊢ ((𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵) → ran (𝐶 ↾ 𝐴) ⊆ ran (𝐶 ↾ 𝐵)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐶 ↾ 𝐴) ⊆ ran (𝐶 ↾ 𝐵)) |
4 | df-ima 4641 | . 2 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴) | |
5 | df-ima 4641 | . 2 ⊢ (𝐶 “ 𝐵) = ran (𝐶 ↾ 𝐵) | |
6 | 3, 4, 5 | 3sstr4g 3200 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3131 ran crn 4629 ↾ cres 4630 “ cima 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-sn 3600 df-pr 3601 df-op 3603 df-br 4006 df-opab 4067 df-xp 4634 df-cnv 4636 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 |
This theorem is referenced by: funimass1 5295 funimass2 5296 fvimacnv 5633 f1imass 5777 ecinxp 6612 sbthlem1 6958 sbthlem2 6959 iscnp4 13757 cnptopco 13761 cnntri 13763 cnrest2 13775 cnptopresti 13777 cnptoprest 13778 metcnp3 14050 |
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