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Mirrors > Home > MPE Home > Th. List > rnmptss | Structured version Visualization version GIF version |
Description: The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
Ref | Expression |
---|---|
rnmptss.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
rnmptss | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmptss.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | fmpt 6289 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶) |
3 | frn 5966 | . 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
4 | 2, 3 | sylbi 206 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ↦ cmpt 4643 ran crn 5039 ⟶wf 5800 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: iunon 7323 iinon 7324 gruiun 9500 smadiadetlem3lem2 20292 tgiun 20594 ustuqtop0 21854 metustss 22166 efabl 24100 efsubm 24101 gsummpt2co 29111 psgnfzto1stlem 29181 locfinreflem 29235 gsumesum 29448 esumlub 29449 esumgect 29479 esum2d 29482 ldgenpisyslem1 29553 sxbrsigalem0 29660 omscl 29684 omsmon 29687 carsgclctunlem2 29708 carsgclctunlem3 29709 pmeasadd 29714 suprnmpt 38350 rnmptssrn 38363 wessf1ornlem 38366 rnmptssd 38380 fourierdlem31 39031 fourierdlem53 39052 fourierdlem111 39110 ioorrnopnlem 39200 saliuncl 39218 salexct3 39236 salgensscntex 39238 sge0rnre 39257 sge0tsms 39273 sge0cl 39274 sge0fsum 39280 sge0sup 39284 sge0gerp 39288 sge0pnffigt 39289 sge0lefi 39291 sge0xaddlem1 39326 sge0xaddlem2 39327 meadjiunlem 39358 meadjiun 39359 |
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