Theorem fnsuppeq0 8191

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 4376	. . 3 ⊢ ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅)
2		un0 4369	. . . . . . . 8 ⊢ (𝐴 ∪ ∅) = 𝐴
3		uncom 4133	. . . . . . . 8 ⊢ (𝐴 ∪ ∅) = (∅ ∪ 𝐴)
4	2, 3	eqtr3i 2760	. . . . . . 7 ⊢ 𝐴 = (∅ ∪ 𝐴)
5	4	fneq2i 6636	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 Fn (∅ ∪ 𝐴))
6	5	biimpi 216	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn (∅ ∪ 𝐴))
7	6	3ad2ant1 1133	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 Fn (∅ ∪ 𝐴))
8		fnex 7209	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊) → 𝐹 ∈ V)
9	8	3adant3 1132	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝐹 ∈ V)
10		simp3 1138	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉)
11		0in 4372	. . . . 5 ⊢ (∅ ∩ 𝐴) = ∅
12	11	a1i 11	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (∅ ∩ 𝐴) = ∅)
13		fnsuppres 8190	. . . 4 ⊢ ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})))
14	7, 9, 10, 12, 13	syl121anc 1377	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})))
15	1, 14	bitr3id 285	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹 ↾ 𝐴) = (𝐴 × {𝑍})))
16		fnresdm 6657	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
17	16	3ad2ant1 1133	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 ↾ 𝐴) = 𝐹)
18	17	eqeq1d 2737	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 ↾ 𝐴) = (𝐴 × {𝑍}) ↔ 𝐹 = (𝐴 × {𝑍})))
19	15, 18	bitrd 279	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308  {csn 4601   × cxp 5652   ↾ cres 5656   Fn wfn 6526  (class class class)co 7405   supp csupp 8159

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-supp 8160

This theorem is referenced by:  fczsupp0  8192  cantnf0  9689  mdegldg  26023  mdeg0  26027  suppovss  32658  fsuppind  42613