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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 7130 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶) |
| 3 | frn 6743 | . 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
| 4 | 2, 3 | sylbi 217 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ↦ cmpt 5225 ran crn 5686 ⟶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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 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-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: mptexw 7977 iunon 8379 iinon 8380 gruiun 10839 subdrgint 20804 smadiadetlem3lem2 22673 tgiun 22986 ustuqtop0 24249 metustss 24564 efabl 26592 efsubm 26593 fnpreimac 32681 prodindf 32848 swrdrn2 32939 gsummpt2co 33051 psgnfzto1stlem 33120 elrgspnsubrunlem1 33251 nsgmgc 33440 nsgqusf1olem1 33441 algextdeglem2 33759 algextdeglem4 33761 locfinreflem 33839 rspectopn 33866 zarcls 33873 zartopn 33874 gsumesum 34060 esumlub 34061 esumgect 34091 esum2d 34094 ldgenpisyslem1 34164 sxbrsigalem0 34273 omscl 34297 omsmon 34300 carsgclctunlem2 34321 carsgclctunlem3 34322 pmeasadd 34327 hgt750lemb 34671 mnurndlem2 44301 suprnmpt 45179 rnmptssrn 45187 wessf1ornlem 45190 rnmptssd 45201 rnmptssbi 45267 liminflelimsuplem 45790 fourierdlem53 46174 fourierdlem111 46232 ioorrnopnlem 46319 salexct3 46357 salgensscntex 46359 sge0rnre 46379 sge0tsms 46395 sge0cl 46396 sge0fsum 46402 sge0sup 46406 sge0gerp 46410 sge0pnffigt 46411 sge0lefi 46413 sge0xaddlem1 46448 sge0xaddlem2 46449 meadjiunlem 46480 |
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