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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 6867 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶) |
3 | frn 6514 | . 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
4 | 2, 3 | sylbi 218 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3138 ⊆ wss 3935 ↦ cmpt 5138 ran crn 5550 ⟶wf 6345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 |
This theorem is referenced by: mptexw 7645 iunon 7967 iinon 7968 gruiun 10210 subdrgint 19513 smadiadetlem3lem2 21206 tgiun 21517 ustuqtop0 22778 metustss 23090 efabl 25061 efsubm 25062 fnpreimac 30345 swrdrn2 30556 gsummpt2co 30614 psgnfzto1stlem 30670 locfinreflem 31004 prodindf 31182 gsumesum 31218 esumlub 31219 esumgect 31249 esum2d 31252 ldgenpisyslem1 31322 sxbrsigalem0 31429 omscl 31453 omsmon 31456 carsgclctunlem2 31477 carsgclctunlem3 31478 pmeasadd 31483 hgt750lemb 31827 mnurndlem2 40498 suprnmpt 41310 rnmptssrn 41322 wessf1ornlem 41325 rnmptssd 41338 rnmptssbi 41414 liminflelimsuplem 41936 fourierdlem31 42304 fourierdlem53 42325 fourierdlem111 42383 ioorrnopnlem 42470 saliuncl 42488 salexct3 42506 salgensscntex 42508 sge0rnre 42527 sge0tsms 42543 sge0cl 42544 sge0fsum 42550 sge0sup 42554 sge0gerp 42558 sge0pnffigt 42559 sge0lefi 42561 sge0xaddlem1 42596 sge0xaddlem2 42597 meadjiunlem 42628 meadjiun 42629 |
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