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Theorem fnsuppeq0 8197
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 4399 . . 3 ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹 supp 𝑍) = ∅)
2 un0 4392 . . . . . . . 8 (𝐴 ∪ ∅) = 𝐴
3 uncom 4150 . . . . . . . 8 (𝐴 ∪ ∅) = (∅ ∪ 𝐴)
42, 3eqtr3i 2755 . . . . . . 7 𝐴 = (∅ ∪ 𝐴)
54fneq2i 6653 . . . . . 6 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
65biimpi 215 . . . . 5 (𝐹 Fn 𝐴𝐹 Fn (∅ ∪ 𝐴))
763ad2ant1 1130 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 Fn (∅ ∪ 𝐴))
8 fnex 7229 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑊) → 𝐹 ∈ V)
983adant3 1129 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝐹 ∈ V)
10 simp3 1135 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → 𝑍𝑉)
11 0in 4395 . . . . 5 (∅ ∩ 𝐴) = ∅
1211a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (∅ ∩ 𝐴) = ∅)
13 fnsuppres 8196 . . . 4 ((𝐹 Fn (∅ ∪ 𝐴) ∧ (𝐹 ∈ V ∧ 𝑍𝑉) ∧ (∅ ∩ 𝐴) = ∅) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
147, 9, 10, 12, 13syl121anc 1372 . . 3 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) ⊆ ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
151, 14bitr3id 284 . 2 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹𝐴) = (𝐴 × {𝑍})))
16 fnresdm 6675 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
17163ad2ant1 1130 . . 3 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → (𝐹𝐴) = 𝐹)
1817eqeq1d 2727 . 2 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹𝐴) = (𝐴 × {𝑍}) ↔ 𝐹 = (𝐴 × {𝑍})))
1915, 18bitrd 278 1 ((𝐹 Fn 𝐴𝐴𝑊𝑍𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3461  cun 3942  cin 3943  wss 3944  c0 4322  {csn 4630   × cxp 5676  cres 5680   Fn wfn 6544  (class class class)co 7419   supp csupp 8165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-supp 8166
This theorem is referenced by:  fczsupp0  8198  cantnf0  9700  mdegldg  26046  mdeg0  26050  suppovss  32547  fsuppind  41955
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