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Theorem fnsuppeq0 7561
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 4169 . . 3 ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅)
2 un0 4163 . . . . . . . 8 (𝐴 ∪ ∅) = 𝐴
3 uncom 3955 . . . . . . . 8 (𝐴 ∪ ∅) = (∅ ∪ 𝐴)
42, 3eqtr3i 2823 . . . . . . 7 𝐴 = (∅ ∪ 𝐴)
54fneq2i 6197 . . . . . 6 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
65biimpi 208 . . . . 5 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
763ad2ant1 1164 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 Fn (∅ ∪ 𝐴))
8 fnex 6710 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑊) → 𝐹 ∈ V)
983adant3 1163 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 ∈ V)
10 simp3 1169 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝑍𝑉)
11 0in 4165 . . . . 5 (∅ ∩ 𝐴) = ∅
1211a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (∅ ∩ 𝐴) = ∅)
13 fnsuppres 7560 . . . 4 ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
147, 9, 10, 12, 13syl121anc 1495 . . 3 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
151, 14syl5bbr 277 . 2 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
16 fnresdm 6211 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
17163ad2ant1 1164 . . 3 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (𝐹𝐴) = 𝐹)
1817eqeq1d 2801 . 2 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹𝐴) = (𝐴 × {𝑍}) ↔ 𝐹 = (𝐴 × {𝑍})))
1915, 18bitrd 271 1 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1108   = wceq 1653  wcel 2157  Vcvv 3385  cun 3767  cin 3768  wss 3769  c0 4115  {csn 4368   × cxp 5310  cres 5314   Fn wfn 6096  (class class class)co 6878   supp csupp 7532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-supp 7533
This theorem is referenced by:  fczsupp0  7562  cantnf0  8822  mdegldg  24167  mdeg0  24171
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