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Theorem fnsuppeq0 7490
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 4114 . . 3 ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅)
2 un0 4108 . . . . . . . 8 (𝐴 ∪ ∅) = 𝐴
3 uncom 3898 . . . . . . . 8 (𝐴 ∪ ∅) = (∅ ∪ 𝐴)
42, 3eqtr3i 2782 . . . . . . 7 𝐴 = (∅ ∪ 𝐴)
54fneq2i 6145 . . . . . 6 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
65biimpi 206 . . . . 5 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
763ad2ant1 1128 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 Fn (∅ ∪ 𝐴))
8 fnex 6643 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑊) → 𝐹 ∈ V)
983adant3 1127 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 ∈ V)
10 simp3 1133 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝑍𝑉)
11 0in 4110 . . . . 5 (∅ ∩ 𝐴) = ∅
1211a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (∅ ∩ 𝐴) = ∅)
13 fnsuppres 7489 . . . 4 ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
147, 9, 10, 12, 13syl121anc 1482 . . 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 6159 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
17163ad2ant1 1128 . . 3 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (𝐹𝐴) = 𝐹)
1817eqeq1d 2760 . 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 1072   = wceq 1630  wcel 2137  Vcvv 3338  cun 3711  cin 3712  wss 3713  c0 4056  {csn 4319   × cxp 5262  cres 5266   Fn wfn 6042  (class class class)co 6811   supp csupp 7461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-supp 7462
This theorem is referenced by:  fczsupp0  7491  cantnf0  8743  mdegldg  24023  mdeg0  24027
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