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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 6743 | . 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 2108 ∀wral 3061 ⊆ wss 3951 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 |
| 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-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 |
| This theorem is referenced by: ffvresb 7145 dffi3 9471 infxpenlem 10053 alephsing 10316 seqexw 14058 srgfcl 20193 mplind 22094 1stckgenlem 23561 psmetxrge0 24323 plyreres 26324 aannenlem1 26370 dfnns2 28362 subuhgr 29303 subupgr 29304 subumgr 29305 subusgr 29306 elrspunidl 33456 rmulccn 33927 esumfsup 34071 sxbrsigalem3 34274 sitgf 34349 ctbssinf 37407 dihf11lem 41268 hdmaprnN 41866 hgmaprnN 41903 ofoafg 43367 naddcnff 43375 ntrrn 44135 mnurndlem1 44300 volicoff 46010 dirkercncflem2 46119 fourierdlem15 46137 fourierdlem42 46164 grimuhgr 47878 |
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