Theorem suppdm 49758

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 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)

Proof of Theorem suppdm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppval1 8191	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍})
2	1	adantr 486	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍})
3		df-nel 3062	. . . . . 6 ⊢ (𝑍 ∉ ran 𝐹 ↔ ¬ 𝑍 ∈ ran 𝐹)
4		fvelrn 7081	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
5	4	3ad2antl1 1204	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
6		eleq1 2848	. . . . . . . . 9 ⊢ (𝑍 = (𝐹‘𝑥) → (𝑍 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
7	6	eqcoms 2768	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑍 → (𝑍 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
8	5, 7	syl5ibrcom 250	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑍 → 𝑍 ∈ ran 𝐹))
9	8	necon3bd 2969	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑍 ∈ ran 𝐹 → (𝐹‘𝑥) ≠ 𝑍))
10	3, 9	biimtrid 245	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑍 ∉ ran 𝐹 → (𝐹‘𝑥) ≠ 𝑍))
11	10	impancom 457	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) ≠ 𝑍))
12	11	ralrimiv 3153	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) ≠ 𝑍)
13		rabid2 3444	. . 3 ⊢ (dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍} ↔ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) ≠ 𝑍)
14	12, 13	sylibr 237	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍})
15	2, 14	eqtr4d 2798	1 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ∉ wnel 3061  ∀wral 3076  {crab 3412  dom cdm 5651  ran crn 5652  Fun wfun 6537  ‘cfv 6543  (class class class)co 7424   supp csupp 8184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-supp 8185

This theorem is used by:  elbigolo1  49805