Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  suppdm Structured version   Visualization version   GIF version

Theorem suppdm 47278
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.)
Assertion
Ref Expression
suppdm (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)

Proof of Theorem suppdm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 suppval1 8154 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
21adantr 479 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
3 df-nel 3045 . . . . . 6 (𝑍 ∉ ran 𝐹 ↔ ¬ 𝑍 ∈ ran 𝐹)
4 fvelrn 7077 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
543ad2antl1 1183 . . . . . . . 8 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
6 eleq1 2819 . . . . . . . . 9 (𝑍 = (𝐹𝑥) → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
76eqcoms 2738 . . . . . . . 8 ((𝐹𝑥) = 𝑍 → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
85, 7syl5ibrcom 246 . . . . . . 7 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑍𝑍 ∈ ran 𝐹))
98necon3bd 2952 . . . . . 6 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑍 ∈ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
103, 9biimtrid 241 . . . . 5 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑍 ∉ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
1110impancom 450 . . . 4 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ≠ 𝑍))
1211ralrimiv 3143 . . 3 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
13 rabid2 3462 . . 3 (dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍} ↔ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
1412, 13sylibr 233 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
152, 14eqtr4d 2773 1 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 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