Theorem suppdm 48472

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 8122	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍})
2	1	adantr 480	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍})
3		df-nel 3030	. . . . . 6 ⊢ (𝑍 ∉ ran 𝐹 ↔ ¬ 𝑍 ∈ ran 𝐹)
4		fvelrn 7030	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
5	4	3ad2antl1 1186	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
6		eleq1 2816	. . . . . . . . 9 ⊢ (𝑍 = (𝐹‘𝑥) → (𝑍 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
7	6	eqcoms 2737	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑍 → (𝑍 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
8	5, 7	syl5ibrcom 247	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑍 → 𝑍 ∈ ran 𝐹))
9	8	necon3bd 2939	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑍 ∈ ran 𝐹 → (𝐹‘𝑥) ≠ 𝑍))
10	3, 9	biimtrid 242	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑍 ∉ ran 𝐹 → (𝐹‘𝑥) ≠ 𝑍))
11	10	impancom 451	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) ≠ 𝑍))
12	11	ralrimiv 3124	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) ≠ 𝑍)
13		rabid2 3436	. . 3 ⊢ (dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍} ↔ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) ≠ 𝑍)
14	12, 13	sylibr 234	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍})
15	2, 14	eqtr4d 2767	1 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∉ wnel 3029  ∀wral 3044  {crab 3402  dom cdm 5631  ran crn 5632  Fun wfun 6493  ‘cfv 6499  (class class class)co 7369   supp csupp 8116

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-supp 8117

This theorem is referenced by:  elbigolo1  48519