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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 2815 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹)) | |
2 | 1 | anbi2d 628 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))) |
3 | fveq2 6884 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
4 | 3 | eleq1d 2812 | . . . 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 7044 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹) | |
7 | vex 3472 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | opeq1 4868 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ⟨𝑦, (𝐹‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩) | |
9 | 8 | eleq1d 2812 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)) |
10 | 7, 9 | spcev 3590 | . . . . 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 6897 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
13 | 12 | elrn2 5885 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ ran 𝐹 ↔ ∃𝑦⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹) |
14 | 11, 13 | sylibr 233 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
15 | 5, 14 | vtoclg 3537 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)) |
16 | 15 | anabsi7 668 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ⟨cop 4629 dom cdm 5669 ran crn 5670 Fun wfun 6530 ‘cfv 6536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-fv 6544 |
This theorem is referenced by: nelrnfvne 7072 fnfvelrn 7075 eldmrexrn 7085 fvn0fvelrnOLD 7156 funfvima 7226 elunirn 7245 funeldmb 7351 rankwflemb 9787 dfac9 10130 fin1a2lem6 10399 gsumpropd2lem 18610 nofv 27541 sltres 27546 nolt02olem 27578 nosupno 27587 noinfno 27602 iedgedg 28814 usgredg3 28977 ushgredgedg 28990 ushgredgedgloop 28992 subgruhgredgd 29045 edginwlk 29397 iedginwlk 29399 opfv 32375 fnpreimac 32401 ccatf1 32618 swrdrn2 32621 zartopn 33385 zarmxt1 33390 bj-elccinfty 36602 bj-minftyccb 36613 icoreunrn 36747 indexdom 37113 diaclN 40432 dia1elN 40436 docaclN 40506 dibclN 40544 sticksstones1 41504 dfac21 42367 harval3 42846 gneispace 43442 cncmpmax 44273 icccncfext 45156 stoweidlem27 45296 stoweidlem29 45298 stoweidlem59 45328 fourierdlem20 45396 fourierdlem63 45438 fourierdlem76 45451 fourierdlem82 45457 fourierdlem93 45468 fourierdlem113 45488 fge0iccico 45639 sge0sn 45648 sge0tsms 45649 sge0cl 45650 sge0isum 45696 hoicvr 45817 funressndmfvrn 46307 fcores 46330 afvelrn 46429 isomushgr 47047 ushrisomgr 47062 suppdm 47447 |
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