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| Mirrors > Home > MPE Home > Th. List > fvelrn | Structured version Visualization version GIF version | ||
| Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.) |
| Ref | Expression |
|---|---|
| fvelrn | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq1 2829 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹)) | |
| 2 | 1 | anbi2d 630 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))) |
| 3 | fveq2 6906 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
| 4 | 3 | eleq1d 2826 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ (𝐹‘𝐴) ∈ ran 𝐹)) |
| 5 | 2, 4 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹))) |
| 6 | funfvop 7070 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) | |
| 7 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 8 | opeq1 4873 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → 〈𝑦, (𝐹‘𝑥)〉 = 〈𝑥, (𝐹‘𝑥)〉) | |
| 9 | 8 | eleq1d 2826 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹 ↔ 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹)) |
| 10 | 7, 9 | spcev 3606 | . . . . 5 ⊢ (〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹 → ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹) |
| 11 | 6, 10 | syl 17 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹) |
| 12 | fvex 6919 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
| 13 | 12 | elrn2 5903 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ ran 𝐹 ↔ ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹) |
| 14 | 11, 13 | sylibr 234 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
| 15 | 5, 14 | vtoclg 3554 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)) |
| 16 | 15 | anabsi7 671 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 〈cop 4632 dom cdm 5685 ran crn 5686 Fun wfun 6555 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: nelrnfvne 7097 fnfvelrn 7100 eldmrexrn 7111 fvn0fvelrnOLD 7183 funfvima 7250 elunirn 7271 funeldmb 7379 rankwflemb 9833 dfac9 10177 fin1a2lem6 10445 gsumpropd2lem 18692 nofv 27702 sltres 27707 nolt02olem 27739 nosupno 27748 noinfno 27763 iedgedg 29067 usgredg3 29233 ushgredgedg 29246 ushgredgedgloop 29248 subgruhgredgd 29301 edginwlk 29653 iedginwlk 29655 cyclnumvtx 29820 opfv 32654 fnpreimac 32681 ccatf1 32933 swrdrn2 32939 zartopn 33874 zarmxt1 33879 bj-elccinfty 37215 bj-minftyccb 37226 icoreunrn 37360 indexdom 37741 diaclN 41052 dia1elN 41056 docaclN 41126 dibclN 41164 sticksstones1 42147 dfac21 43078 harval3 43551 gneispace 44147 cncmpmax 45037 icccncfext 45902 stoweidlem27 46042 stoweidlem29 46044 stoweidlem59 46074 fourierdlem20 46142 fourierdlem63 46184 fourierdlem76 46197 fourierdlem82 46203 fourierdlem93 46214 fourierdlem113 46234 fge0iccico 46385 sge0sn 46394 sge0tsms 46395 sge0cl 46396 sge0isum 46442 hoicvr 46563 funressndmfvrn 47056 fcores 47079 afvelrn 47180 isubgredg 47852 gricushgr 47886 ushggricedg 47896 suppdm 48427 |
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