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Mirrors > Home > ILE Home > Th. List > ffvelrn | GIF version |
Description: A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.) |
Ref | Expression |
---|---|
ffvelrn | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5280 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fnfvelrn 5560 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ ran 𝐹) | |
3 | 1, 2 | sylan 281 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ ran 𝐹) |
4 | frn 5289 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
5 | 4 | sseld 3101 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹‘𝐶) ∈ ran 𝐹 → (𝐹‘𝐶) ∈ 𝐵)) |
6 | 5 | adantr 274 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) ∈ ran 𝐹 → (𝐹‘𝐶) ∈ 𝐵)) |
7 | 3, 6 | mpd 13 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 ran crn 4548 Fn wfn 5126 ⟶wf 5127 ‘cfv 5131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 |
This theorem is referenced by: ffvelrni 5562 ffvelrnda 5563 dffo3 5575 ffnfv 5586 ffvresb 5591 fcompt 5598 fsn2 5602 fvconst 5616 foco2 5663 fcofo 5693 cocan1 5696 isocnv 5720 isores2 5722 isopolem 5731 isosolem 5733 fovrn 5921 off 6002 mapsncnv 6597 2dom 6707 enm 6722 xpdom2 6733 xpmapenlem 6751 fiintim 6825 isotilem 6901 updjudhf 6972 exmidomniim 7021 shftf 10634 summodclem2a 11182 isumcl 11226 mertenslem2 11337 nn0seqcvgd 11758 algrf 11762 eucalg 11776 phimullem 11937 upxp 12480 uptx 12482 txhmeo 12527 cncfmet 12787 dvaddxxbr 12873 dvcj 12881 dvfre 12882 |
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