Theorem fnsuppeq0 7979

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 4328	. . 3 ⊢ ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅)
2		un0 4321	. . . . . . . 8 ⊢ (𝐴 ∪ ∅) = 𝐴
3		uncom 4083	. . . . . . . 8 ⊢ (𝐴 ∪ ∅) = (∅ ∪ 𝐴)
4	2, 3	eqtr3i 2768	. . . . . . 7 ⊢ 𝐴 = (∅ ∪ 𝐴)
5	4	fneq2i 6515	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 Fn (∅ ∪ 𝐴))
6	5	biimpi 215	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn (∅ ∪ 𝐴))
7	6	3ad2ant1 1131	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 Fn (∅ ∪ 𝐴))
8		fnex 7075	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊) → 𝐹 ∈ V)
9	8	3adant3 1130	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 ∈ V)
10		simp3 1136	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉)
11		0in 4324	. . . . 5 ⊢ (∅ ∩ 𝐴) = ∅
12	11	a1i 11	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (∅ ∩ 𝐴) = ∅)
13		fnsuppres 7978	. . . 4 ⊢ ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})))
14	7, 9, 10, 12, 13	syl121anc 1373	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})))
15	1, 14	bitr3id 284	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})))
16		fnresdm 6535	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
17	16	3ad2ant1 1131	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 ↾ 𝐴) = 𝐹)
18	17	eqeq1d 2740	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝐴) = (𝐴 × {𝑍}) ↔ 𝐹 = (𝐴 × {𝑍})))
19	15, 18	bitrd 278	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558   × cxp 5578   ↾ cres 5582   Fn wfn 6413  (class class class)co 7255   supp csupp 7948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-supp 7949

This theorem is referenced by:  fczsupp0  7980  cantnf0  9363  mdegldg  25136  mdeg0  25140  suppovss  30919  fsuppind  40202