![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2817 | . . . . 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 6902 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
4 | 3 | eleq1d 2814 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ (𝐹‘𝐴) ∈ ran 𝐹)) |
5 | 2, 4 | imbi12d 343 | . . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹))) |
6 | funfvop 7064 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹) | |
7 | vex 3477 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | opeq1 4878 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ⟨𝑦, (𝐹‘𝑥)⟩ = ⟨𝑥, (𝐹‘𝑥)⟩) | |
9 | 8 | eleq1d 2814 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (⟨𝑦, (𝐹‘𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)) |
10 | 7, 9 | spcev 3595 | . . . . 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 6915 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
13 | 12 | elrn2 5899 | . . . 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 3542 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)) |
16 | 15 | anabsi7 669 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ⟨cop 4638 dom cdm 5682 ran crn 5683 Fun wfun 6547 ‘cfv 6553 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 |
This theorem is referenced by: nelrnfvne 7092 fnfvelrn 7095 eldmrexrn 7106 fvn0fvelrnOLD 7178 funfvima 7248 elunirn 7267 funeldmb 7373 rankwflemb 9824 dfac9 10167 fin1a2lem6 10436 gsumpropd2lem 18646 nofv 27610 sltres 27615 nolt02olem 27647 nosupno 27656 noinfno 27671 iedgedg 28883 usgredg3 29049 ushgredgedg 29062 ushgredgedgloop 29064 subgruhgredgd 29117 edginwlk 29469 iedginwlk 29471 opfv 32452 fnpreimac 32478 ccatf1 32693 swrdrn2 32696 zartopn 33509 zarmxt1 33514 bj-elccinfty 36726 bj-minftyccb 36737 icoreunrn 36871 indexdom 37240 diaclN 40555 dia1elN 40559 docaclN 40629 dibclN 40667 sticksstones1 41650 dfac21 42521 harval3 42999 gneispace 43595 cncmpmax 44425 icccncfext 45304 stoweidlem27 45444 stoweidlem29 45446 stoweidlem59 45476 fourierdlem20 45544 fourierdlem63 45586 fourierdlem76 45599 fourierdlem82 45605 fourierdlem93 45616 fourierdlem113 45636 fge0iccico 45787 sge0sn 45796 sge0tsms 45797 sge0cl 45798 sge0isum 45844 hoicvr 45965 funressndmfvrn 46455 fcores 46478 afvelrn 46577 gricushgr 47261 ushggricedg 47271 suppdm 47656 |
Copyright terms: Public domain | W3C validator |