Theorem suppdm 49132

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 8146	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍})
2	1	adantr 484	. 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 7057	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
5	4	3ad2antl1 1199	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
6		eleq1 2850	. . . . . . . . 9 ⊢ (𝑍 = (𝐹‘𝑥) → (𝑍 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
7	6	eqcoms 2770	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑍 → (𝑍 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
8	5, 7	syl5ibrcom 249	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑍 → 𝑍 ∈ ran 𝐹))
9	8	necon3bd 2971	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (¬ 𝑍 ∈ ran 𝐹 → (𝐹‘𝑥) ≠ 𝑍))
10	3, 9	biimtrid 244	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑥 ∈ dom 𝐹) → (𝑍 ∉ ran 𝐹 → (𝐹‘𝑥) ≠ 𝑍))
11	10	impancom 455	. . . 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 3447	. . 3 ⊢ (dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍} ↔ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) ≠ 𝑍)
14	12, 13	sylibr 236	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → dom 𝐹 = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ≠ 𝑍})
15	2, 14	eqtr4d 2800	1 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957   ∉ wnel 3061  ∀wral 3076  {crab 3414  dom cdm 5647  ran crn 5648  Fun wfun 6515  ‘cfv 6521  (class class class)co 7396   supp csupp 8140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-supp 8141

This theorem is referenced by:  elbigolo1  49179