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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 6859 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | frn 6493 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
3 | 1, 2 | sylbir 238 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 ran crn 5520 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 |
This theorem is referenced by: ffvresb 6865 dffi3 8879 infxpenlem 9424 alephsing 9687 seqexw 13380 srgfcl 19258 mplind 20741 1stckgenlem 22158 psmetxrge0 22920 plyreres 24879 aannenlem1 24924 subuhgr 27076 subupgr 27077 subumgr 27078 subusgr 27079 elrspunidl 31014 rmulccn 31281 esumfsup 31439 sxbrsigalem3 31640 sitgf 31715 ctbssinf 34823 dihf11lem 38562 hdmaprnN 39160 hgmaprnN 39197 ntrrn 40825 mnurndlem1 40989 volicoff 42637 dirkercncflem2 42746 fourierdlem15 42764 fourierdlem42 42791 |
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