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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 5644 | . 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 2148 ⟶wf 5207 ‘cfv 5211 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 |
This theorem is referenced by: omgadd 10753 cjcl 10828 climmpt 11279 cn1lem 11293 climcn1lem 11298 fsumrelem 11450 efcl 11643 sincl 11685 coscl 11686 algcvg 12018 algcvgb 12020 algcvga 12021 algfx 12022 eucalgcvga 12028 eucalg 12029 sqpweven 12145 2sqpwodd 12146 ennnfonelemnn0 12393 relogcl 13916 nninfomnilem 14390 |
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