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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 5046 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fnfvelrn 5299 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ ran 𝐹) | |
3 | 1, 2 | sylan 267 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ ran 𝐹) |
4 | frn 5052 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
5 | 4 | sseld 2944 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹‘𝐶) ∈ ran 𝐹 → (𝐹‘𝐶) ∈ 𝐵)) |
6 | 5 | adantr 261 | . 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 97 ∈ wcel 1393 ran crn 4346 Fn wfn 4897 ⟶wf 4898 ‘cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 |
This theorem is referenced by: ffvelrni 5301 ffvelrnda 5302 dffo3 5314 foco2 5318 ffnfv 5323 ffvresb 5328 fcompt 5333 fsn2 5337 fvconst 5351 fcofo 5424 cocan1 5427 isocnv 5451 isores2 5453 isopolem 5461 isosolem 5463 fovrn 5643 off 5724 2dom 6285 enm 6294 xpdom2 6305 shftf 9431 nn0seqcvgd 9880 |
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