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Mirrors > Home > ILE Home > Th. List > fnfvelrn | GIF version |
Description: A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.) |
Ref | Expression |
---|---|
fnfvelrn | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvelrn 5469 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐹‘𝐵) ∈ ran 𝐹) | |
2 | 1 | funfni 5148 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ∈ ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1445 ran crn 4468 Fn wfn 5044 ‘cfv 5049 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-iota 5014 df-fun 5051 df-fn 5052 df-fv 5057 |
This theorem is referenced by: ffvelrn 5471 fnovrn 5830 fo1stresm 5970 fo2ndresm 5971 fo2ndf 6030 phplem4 6651 phplem4on 6663 frec2uzrand 9961 frecuzrdglem 9967 frecuzrdg0 9969 frecuzrdg0t 9978 uzin2 10535 |
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