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Mirrors > Home > ILE Home > Th. List > ffvelcdmi | GIF version |
Description: A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.) |
Ref | Expression |
---|---|
ffvelcdmi.1 | ⊢ 𝐹:𝐴⟶𝐵 |
Ref | Expression |
---|---|
ffvelcdmi | ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffvelcdmi.1 | . 2 ⊢ 𝐹:𝐴⟶𝐵 | |
2 | ffvelcdm 5691 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) | |
3 | 1, 2 | mpan 424 | 1 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 ⟶wf 5250 ‘cfv 5254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-sbc 2986 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 |
This theorem is referenced by: omgadd 10873 cjcl 10992 climmpt 11443 cn1lem 11457 climcn1lem 11462 fsumrelem 11614 efcl 11807 sincl 11849 coscl 11850 algcvg 12186 algcvgb 12188 algcvga 12189 algfx 12190 eucalgcvga 12196 eucalg 12197 sqpweven 12313 2sqpwodd 12314 ennnfonelemnn0 12579 relogcl 14997 nninfomnilem 15508 |
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