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 42969
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 7503 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
21adantr 472 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
3 df-nel 3041 . . . . . 6 (𝑍 ∉ ran 𝐹 ↔ ¬ 𝑍 ∈ ran 𝐹)
4 fvelrn 6542 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
543ad2antl1 1236 . . . . . . . 8 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
6 eleq1 2832 . . . . . . . . 9 (𝑍 = (𝐹𝑥) → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
76eqcoms 2773 . . . . . . . 8 ((𝐹𝑥) = 𝑍 → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
85, 7syl5ibrcom 238 . . . . . . 7 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑍𝑍 ∈ ran 𝐹))
98necon3bd 2951 . . . . . 6 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑍 ∈ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
103, 9syl5bi 233 . . . . 5 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑍 ∉ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
1110impancom 443 . . . 4 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ≠ 𝑍))
1211ralrimiv 3112 . . 3 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
13 rabid2 3266 . . 3 (dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍} ↔ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
1412, 13sylibr 225 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
152, 14eqtr4d 2802 1 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wnel 3040  wral 3055  {crab 3059  dom cdm 5277  ran crn 5278  Fun wfun 6062  cfv 6068  (class class class)co 6842   supp csupp 7497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-supp 7498
This theorem is referenced by:  elbigolo1  43020
  Copyright terms: Public domain W3C validator