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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 4989 | . . 3 ⊢ (𝐹:A⟶B → 𝐹 Fn A) | |
2 | fnfvelrn 5242 | . . 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 4995 | . . . 4 ⊢ (𝐹:A⟶B → ran 𝐹 ⊆ B) | |
5 | 4 | sseld 2938 | . . 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 1390 ran crn 4289 Fn wfn 4840 ⟶wf 4841 ‘cfv 4845 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 |
This theorem is referenced by: ffvelrni 5244 ffvelrnda 5245 dffo3 5257 foco2 5261 ffnfv 5266 ffvresb 5271 fcompt 5276 fsn2 5280 fvconst 5294 fcofo 5367 cocan1 5370 isocnv 5394 isores2 5396 isopolem 5404 isosolem 5406 fovrn 5585 off 5666 2dom 6221 enm 6230 xpdom2 6241 |
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