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| Mirrors > Home > MPE Home > Th. List > fimass | Structured version Visualization version GIF version | ||
| Description: The image of a class under a function with domain and codomain is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| fimass | ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imassrn 6089 | . 2 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹 | |
| 2 | frn 6743 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
| 3 | 1, 2 | sstrid 3995 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ⊆ wss 3951 ran crn 5686 “ cima 5688 ⟶wf 6557 |
| 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 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-ral 3062 df-rex 3071 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 df-f 6565 |
| This theorem is referenced by: fimassd 6757 fimarab 6983 f1imaen2g 9055 domunsncan 9112 fissuni 9397 fipreima 9398 carduniima 10136 psgnunilem1 19511 fbasrn 23892 imaelfm 23959 wlkres 29688 trlreslem 29717 tocyccntz 33164 rhmimaidl 33460 nummin 35105 hashscontpowcl 42121 relpfrlem 44974 limsupvaluz 45723 fundcmpsurbijinjpreimafv 47394 fundcmpsurinjimaid 47398 |
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