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Theorem suppdm 45433
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 7875 . . 3 ((Fun 𝐹𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
21adantr 484 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
3 df-nel 3040 . . . . . 6 (𝑍 ∉ ran 𝐹 ↔ ¬ 𝑍 ∈ ran 𝐹)
4 fvelrn 6867 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
543ad2antl1 1186 . . . . . . . 8 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
6 eleq1 2821 . . . . . . . . 9 (𝑍 = (𝐹𝑥) → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
76eqcoms 2747 . . . . . . . 8 ((𝐹𝑥) = 𝑍 → (𝑍 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
85, 7syl5ibrcom 250 . . . . . . 7 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑍𝑍 ∈ ran 𝐹))
98necon3bd 2949 . . . . . 6 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑍 ∈ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
103, 9syl5bi 245 . . . . 5 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑍 ∉ ran 𝐹 → (𝐹𝑥) ≠ 𝑍))
1110impancom 455 . . . 4 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ≠ 𝑍))
1211ralrimiv 3096 . . 3 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
13 rabid2 3285 . . 3 (dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍} ↔ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) ≠ 𝑍)
1412, 13sylibr 237 . 2 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ≠ 𝑍})
152, 14eqtr4d 2777 1 (((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2935  wnel 3039  wral 3054  {crab 3058  dom cdm 5535  ran crn 5536  Fun wfun 6344  cfv 6350  (class class class)co 7183   supp csupp 7869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-supp 7870
This theorem is referenced by:  elbigolo1  45485
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