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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 7102 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | frn 6711 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
3 | 1, 2 | sylbir 234 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3060 ⊆ wss 3944 ran crn 5670 Fn wfn 6527 ⟶wf 6528 ‘cfv 6532 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 |
This theorem is referenced by: ffvresb 7108 dffi3 9408 infxpenlem 9990 alephsing 10253 seqexw 13964 srgfcl 19977 mplind 21560 1stckgenlem 22986 psmetxrge0 23748 plyreres 25725 aannenlem1 25770 subuhgr 28408 subupgr 28409 subumgr 28410 subusgr 28411 elrspunidl 32397 rmulccn 32739 esumfsup 32899 sxbrsigalem3 33102 sitgf 33177 ctbssinf 36091 dihf11lem 39942 hdmaprnN 40540 hgmaprnN 40577 ofoafg 41875 naddcnff 41883 ntrrn 42644 mnurndlem1 42811 volicoff 44484 dirkercncflem2 44593 fourierdlem15 44611 fourierdlem42 44638 |
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