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Mirrors > Home > ILE Home > Th. List > ffvelcdm | GIF version |
Description: A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.) |
Ref | Expression |
---|---|
ffvelcdm | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5360 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fnfvelrn 5643 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ ran 𝐹) | |
3 | 1, 2 | sylan 283 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ ran 𝐹) |
4 | frn 5369 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
5 | 4 | sseld 3154 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹‘𝐶) ∈ ran 𝐹 → (𝐹‘𝐶) ∈ 𝐵)) |
6 | 5 | adantr 276 | . 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 104 ∈ wcel 2148 ran crn 4623 Fn wfn 5206 ⟶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: ffvelcdmi 5645 ffvelcdmda 5646 dffo3 5658 ffnfv 5669 ffvresb 5674 fcompt 5681 fsn2 5685 fvconst 5699 foco2 5748 fcofo 5778 cocan1 5781 isocnv 5805 isores2 5807 isopolem 5816 isosolem 5818 fovcdm 6010 off 6088 mapsncnv 6688 2dom 6798 enm 6813 xpdom2 6824 xpmapenlem 6842 fiintim 6921 isotilem 6998 updjudhf 7071 exmidomniim 7132 shftf 10810 summodclem2a 11360 isumcl 11404 mertenslem2 11515 nn0seqcvgd 12011 algrf 12015 eucalg 12029 phimullem 12195 pcmpt 12311 pcprod 12314 mhmpropd 12734 upxp 13405 uptx 13407 txhmeo 13452 cncfmet 13712 dvaddxxbr 13798 dvcj 13806 dvfre 13807 lgsdir 14069 lgsdi 14071 bj-charfunr 14184 |
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