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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 2893 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹)) | |
2 | 1 | anbi2d 624 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))) |
3 | fveq2 6432 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
4 | 3 | eleq1d 2890 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ (𝐹‘𝐴) ∈ ran 𝐹)) |
5 | 2, 4 | imbi12d 336 | . . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹))) |
6 | funfvop 6577 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) | |
7 | vex 3416 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | opeq1 4622 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → 〈𝑦, (𝐹‘𝑥)〉 = 〈𝑥, (𝐹‘𝑥)〉) | |
9 | 8 | eleq1d 2890 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹 ↔ 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹)) |
10 | 7, 9 | spcev 3516 | . . . . 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 6445 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
13 | 12 | elrn2 5597 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ ran 𝐹 ↔ ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹) |
14 | 11, 13 | sylibr 226 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
15 | 5, 14 | vtoclg 3481 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)) |
16 | 15 | anabsi7 663 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∃wex 1880 ∈ wcel 2166 〈cop 4402 dom cdm 5341 ran crn 5342 Fun wfun 6116 ‘cfv 6122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-iota 6085 df-fun 6124 df-fn 6125 df-fv 6130 |
This theorem is referenced by: nelrnfvne 6601 fnfvelrn 6604 eldmrexrn 6613 fvn0fvelrn 6680 funfvima 6747 elunirn 6763 rankwflemb 8932 dfac9 9272 fin1a2lem6 9541 gsumpropd2lem 17625 iedgedg 26347 usgredg3 26511 ushgredgedg 26524 ushgredgedgloop 26526 ushgredgedgloopOLD 26527 subgruhgredgd 26580 edginwlk 26931 iedginwlk 26933 opfv 29996 funeldmb 32202 nofv 32348 sltres 32353 nolt02olem 32382 nosupno 32387 bj-elccinfty 33640 bj-minftyccb 33651 icoreunrn 33751 indexdom 34071 diaclN 37124 dia1elN 37128 docaclN 37198 dibclN 37236 dfac21 38478 gneispace 39271 cncmpmax 40008 icccncfext 40894 stoweidlem27 41037 stoweidlem29 41039 stoweidlem59 41069 fourierdlem20 41137 fourierdlem63 41179 fourierdlem76 41192 fourierdlem82 41198 fourierdlem93 41209 fourierdlem113 41229 fge0iccico 41377 sge0sn 41386 sge0tsms 41387 sge0cl 41388 sge0isum 41434 hoicvr 41555 funressndmfvrn 41974 afvelrn 42069 isomushgr 42543 ushrisomgr 42558 suppdm 43146 |
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