Theorem fnsuppeq0 7368

Description: The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.)

Assertion

Ref	Expression
fnsuppeq0	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))

Proof of Theorem fnsuppeq0

Step	Hyp	Ref	Expression
1		ss0b 4006	. . 3 ⊢ ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅)
2		un0 4000	. . . . . . . 8 ⊢ (𝐴 ∪ ∅) = 𝐴
3		uncom 3790	. . . . . . . 8 ⊢ (𝐴 ∪ ∅) = (∅ ∪ 𝐴)
4	2, 3	eqtr3i 2675	. . . . . . 7 ⊢ 𝐴 = (∅ ∪ 𝐴)
5	4	fneq2i 6024	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 Fn (∅ ∪ 𝐴))
6	5	biimpi 206	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn (∅ ∪ 𝐴))
7	6	3ad2ant1 1102	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 Fn (∅ ∪ 𝐴))
8		fnex 6522	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊) → 𝐹 ∈ V)
9	8	3adant3 1101	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 ∈ V)
10		simp3 1083	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉)
11		0in 4002	. . . . 5 ⊢ (∅ ∩ 𝐴) = ∅
12	11	a1i 11	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (∅ ∩ 𝐴) = ∅)
13		fnsuppres 7367	. . . 4 ⊢ ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})))
14	7, 9, 10, 12, 13	syl121anc 1371	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})))
15	1, 14	syl5bbr 274	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})))
16		fnresdm 6038	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
17	16	3ad2ant1 1102	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 ↾ 𝐴) = 𝐹)
18	17	eqeq1d 2653	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝐴) = (𝐴 × {𝑍}) ↔ 𝐹 = (𝐴 × {𝑍})))
19	15, 18	bitrd 268	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {csn 4210   × cxp 5141   ↾ cres 5145   Fn wfn 5921  (class class class)co 6690   supp csupp 7340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-supp 7341

This theorem is referenced by:  fczsupp0  7369  cantnf0  8610  mdegldg  23871  mdeg0  23875