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Theorem suppdm 42065
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 7286 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
21adantr 481 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
3 df-nel 2895 . . . . . 6 (𝑍 ∉ ran 𝐹 ↔ ¬ 𝑍 ∈ ran 𝐹)
4 fvelrn 6338 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
543ad2antl1 1221 . . . . . . . 8 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
6 eleq1 2687 . . . . . . . . 9 (𝑍 = (𝐹𝑥) → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
76eqcoms 2628 . . . . . . . 8 ((𝐹𝑥) = 𝑍 → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
85, 7syl5ibrcom 237 . . . . . . 7 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑍𝑍 ∈ ran 𝐹))
98necon3bd 2805 . . . . . 6 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑍 ∈ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
103, 9syl5bi 232 . . . . 5 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑍 ∉ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
1110impancom 456 . . . 4 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ≠ 𝑍))
1211ralrimiv 2962 . . 3 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
13 rabid2 3113 . . 3 (dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍} ↔ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
1412, 13sylibr 224 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
152, 14eqtr4d 2657 1 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wnel 2894  wral 2909  {crab 2913  dom cdm 5104  ran crn 5105  Fun wfun 5870  cfv 5876  (class class class)co 6635   supp csupp 7280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-supp 7281
This theorem is referenced by:  elbigolo1  42116
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