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Theorem fnsuppeq0 8217
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 4401 . . 3 ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅)
2 un0 4394 . . . . . . . 8 (𝐴 ∪ ∅) = 𝐴
3 uncom 4158 . . . . . . . 8 (𝐴 ∪ ∅) = (∅ ∪ 𝐴)
42, 3eqtr3i 2767 . . . . . . 7 𝐴 = (∅ ∪ 𝐴)
54fneq2i 6666 . . . . . 6 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
65biimpi 216 . . . . 5 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
763ad2ant1 1134 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 Fn (∅ ∪ 𝐴))
8 fnex 7237 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑊) → 𝐹 ∈ V)
983adant3 1133 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 ∈ V)
10 simp3 1139 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝑍𝑉)
11 0in 4397 . . . . 5 (∅ ∩ 𝐴) = ∅
1211a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (∅ ∩ 𝐴) = ∅)
13 fnsuppres 8216 . . . 4 ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
147, 9, 10, 12, 13syl121anc 1377 . . 3 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
151, 14bitr3id 285 . 2 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
16 fnresdm 6687 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
17163ad2ant1 1134 . . 3 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (𝐹𝐴) = 𝐹)
1817eqeq1d 2739 . 2 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹𝐴) = (𝐴 × {𝑍}) ↔ 𝐹 = (𝐴 × {𝑍})))
1915, 18bitrd 279 1 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626   × cxp 5683  cres 5687   Fn wfn 6556  (class class class)co 7431   supp csupp 8185
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  fczsupp0  8218  cantnf0  9715  mdegldg  26105  mdeg0  26109  suppovss  32690  fsuppind  42600
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