![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6091 | . 2 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹 | |
2 | frn 6744 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
3 | 1, 2 | sstrid 4007 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3963 ran crn 5690 “ cima 5692 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-f 6567 |
This theorem is referenced by: fimassd 6758 fimarab 6983 f1imaen2g 9054 domunsncan 9111 fissuni 9395 fipreima 9396 carduniima 10134 psgnunilem1 19526 fbasrn 23908 imaelfm 23975 wlkres 29703 trlreslem 29732 tocyccntz 33147 rhmimaidl 33440 nummin 35084 hashscontpowcl 42102 limsupvaluz 45664 fundcmpsurbijinjpreimafv 47332 fundcmpsurinjimaid 47336 |
Copyright terms: Public domain | W3C validator |