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Mirrors > Home > MPE Home > Th. List > fnfvrnss | Structured version Visualization version GIF version |
Description: An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.) |
Ref | Expression |
---|---|
fnfvrnss | ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffnfv 7139 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | frn 6744 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
3 | 1, 2 | sylbir 235 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 ran crn 5690 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 |
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-10 2139 ax-11 2155 ax-12 2175 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-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 |
This theorem is referenced by: ffvresb 7145 dffi3 9469 infxpenlem 10051 alephsing 10314 seqexw 14055 srgfcl 20214 mplind 22112 1stckgenlem 23577 psmetxrge0 24339 plyreres 26339 aannenlem1 26385 dfnns2 28377 subuhgr 29318 subupgr 29319 subumgr 29320 subusgr 29321 elrspunidl 33436 rmulccn 33889 esumfsup 34051 sxbrsigalem3 34254 sitgf 34329 ctbssinf 37389 dihf11lem 41249 hdmaprnN 41847 hgmaprnN 41884 ofoafg 43344 naddcnff 43352 ntrrn 44112 mnurndlem1 44277 volicoff 45951 dirkercncflem2 46060 fourierdlem15 46078 fourierdlem42 46105 grimuhgr 47816 |
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