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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 2827 | . . . . 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 6907 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
4 | 3 | eleq1d 2824 | . . . 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 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | opeq1 4878 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → 〈𝑦, (𝐹‘𝑥)〉 = 〈𝑥, (𝐹‘𝑥)〉) | |
9 | 8 | eleq1d 2824 | . . . . . 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 6920 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
13 | 12 | elrn2 5906 | . . . 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 1537 ∃wex 1776 ∈ wcel 2106 〈cop 4637 dom cdm 5689 ran crn 5690 Fun wfun 6557 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: nelrnfvne 7097 fnfvelrn 7100 eldmrexrn 7111 fvn0fvelrnOLD 7183 funfvima 7250 elunirn 7271 funeldmb 7379 rankwflemb 9831 dfac9 10175 fin1a2lem6 10443 gsumpropd2lem 18705 nofv 27717 sltres 27722 nolt02olem 27754 nosupno 27763 noinfno 27778 iedgedg 29082 usgredg3 29248 ushgredgedg 29261 ushgredgedgloop 29263 subgruhgredgd 29316 edginwlk 29668 iedginwlk 29670 opfv 32661 fnpreimac 32688 ccatf1 32918 swrdrn2 32924 zartopn 33836 zarmxt1 33841 bj-elccinfty 37197 bj-minftyccb 37208 icoreunrn 37342 indexdom 37721 diaclN 41033 dia1elN 41037 docaclN 41107 dibclN 41145 sticksstones1 42128 dfac21 43055 harval3 43528 gneispace 44124 cncmpmax 44970 icccncfext 45843 stoweidlem27 45983 stoweidlem29 45985 stoweidlem59 46015 fourierdlem20 46083 fourierdlem63 46125 fourierdlem76 46138 fourierdlem82 46144 fourierdlem93 46155 fourierdlem113 46175 fge0iccico 46326 sge0sn 46335 sge0tsms 46336 sge0cl 46337 sge0isum 46383 hoicvr 46504 funressndmfvrn 46994 fcores 47017 afvelrn 47118 isubgredg 47790 gricushgr 47824 ushggricedg 47834 suppdm 48356 |
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