![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7129 | . 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 1536 ∈ wcel 2105 ∀wral 3058 ⊆ wss 3962 ↦ cmpt 5230 ran crn 5689 ⟶wf 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-fun 6564 df-fn 6565 df-f 6566 |
This theorem is referenced by: mptexw 7975 iunon 8377 iinon 8378 gruiun 10836 subdrgint 20820 smadiadetlem3lem2 22688 tgiun 23001 ustuqtop0 24264 metustss 24579 efabl 26606 efsubm 26607 fnpreimac 32687 swrdrn2 32923 gsummpt2co 33033 psgnfzto1stlem 33102 nsgmgc 33419 nsgqusf1olem1 33420 algextdeglem2 33723 algextdeglem4 33725 locfinreflem 33800 rspectopn 33827 zarcls 33834 zartopn 33835 prodindf 34003 gsumesum 34039 esumlub 34040 esumgect 34070 esum2d 34073 ldgenpisyslem1 34143 sxbrsigalem0 34252 omscl 34276 omsmon 34279 carsgclctunlem2 34300 carsgclctunlem3 34301 pmeasadd 34306 hgt750lemb 34649 mnurndlem2 44277 suprnmpt 45116 rnmptssrn 45124 wessf1ornlem 45127 rnmptssd 45138 rnmptssbi 45205 liminflelimsuplem 45730 fourierdlem53 46114 fourierdlem111 46172 ioorrnopnlem 46259 salexct3 46297 salgensscntex 46299 sge0rnre 46319 sge0tsms 46335 sge0cl 46336 sge0fsum 46342 sge0sup 46346 sge0gerp 46350 sge0pnffigt 46351 sge0lefi 46353 sge0xaddlem1 46388 sge0xaddlem2 46389 meadjiunlem 46420 |
Copyright terms: Public domain | W3C validator |