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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 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))

Proof of Theorem fnsuppeq0
StepHypRef Expression
1 ss0b 4006 . . 3 ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅)
2 un0 4000 . . . . . . . 8 (𝐴 ∪ ∅) = 𝐴
3 uncom 3790 . . . . . . . 8 (𝐴 ∪ ∅) = (∅ ∪ 𝐴)
42, 3eqtr3i 2675 . . . . . . 7 𝐴 = (∅ ∪ 𝐴)
54fneq2i 6024 . . . . . 6 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
65biimpi 206 . . . . 5 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
763ad2ant1 1102 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 Fn (∅ ∪ 𝐴))
8 fnex 6522 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑊) → 𝐹 ∈ V)
983adant3 1101 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 ∈ V)
10 simp3 1083 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝑍𝑉)
11 0in 4002 . . . . 5 (∅ ∩ 𝐴) = ∅
1211a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (∅ ∩ 𝐴) = ∅)
13 fnsuppres 7367 . . . 4 ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
147, 9, 10, 12, 13syl121anc 1371 . . 3 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
151, 14syl5bbr 274 . 2 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
16 fnresdm 6038 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
17163ad2ant1 1102 . . 3 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (𝐹𝐴) = 𝐹)
1817eqeq1d 2653 . 2 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹𝐴) = (𝐴 × {𝑍}) ↔ 𝐹 = (𝐴 × {𝑍})))
1915, 18bitrd 268 1 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))
Colors of variables: wff setvar class
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
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