| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > imass2d | Structured version Visualization version GIF version | ||
| Description: Subset theorem for image. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| Ref | Expression |
|---|---|
| imass2d.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| Ref | Expression |
|---|---|
| imass2d | ⊢ (𝜑 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imass2d.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 2 | imass2 6120 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ⊆ wss 3951 “ cima 5688 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: liminflelimsuplem 45790 |
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