![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > suppdm | Structured version Visualization version GIF version |
Description: If the range of a function does not contain the zero, the support of the function equals its domain. (Contributed by AV, 20-May-2020.) |
Ref | Expression |
---|---|
suppdm | ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppval1 8154 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍}) | |
2 | 1 | adantr 479 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍}) |
3 | df-nel 3045 | . . . . . 6 ⊢ (𝑍 ∉ ran 𝐹 ↔ ¬ 𝑍 ∈ ran 𝐹) | |
4 | fvelrn 7077 | . . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) | |
5 | 4 | 3ad2antl1 1183 | . . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
6 | eleq1 2819 | . . . . . . . . 9 ⊢ (𝑍 = (𝐹‘𝑥) → (𝑍 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹)) | |
7 | 6 | eqcoms 2738 | . . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑍 → (𝑍 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹)) |
8 | 5, 7 | syl5ibrcom 246 | . . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑍 → 𝑍 ∈ ran 𝐹)) |
9 | 8 | necon3bd 2952 | . . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑍 ∈ ran 𝐹 → (𝐹‘𝑥) ≠ 𝑍)) |
10 | 3, 9 | biimtrid 241 | . . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑍 ∉ ran 𝐹 → (𝐹‘𝑥) ≠ 𝑍)) |
11 | 10 | impancom 450 | . . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) ≠ 𝑍)) |
12 | 11 | ralrimiv 3143 | . . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) ≠ 𝑍) |
13 | rabid2 3462 | . . 3 ⊢ (dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍} ↔ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) ≠ 𝑍) | |
14 | 12, 13 | sylibr 233 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍}) |
15 | 2, 14 | eqtr4d 2773 | 1 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∉ wnel 3044 ∀wral 3059 {crab 3430 dom cdm 5675 ran crn 5676 Fun wfun 6536 ‘cfv 6542 (class class class)co 7411 supp csupp 8148 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-supp 8149 |
This theorem is referenced by: elbigolo1 47330 |
Copyright terms: Public domain | W3C validator |