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Theorem fvn0elsuppb 7997
Description: The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.)
Assertion
Ref Expression
fvn0elsuppb ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅)))

Proof of Theorem fvn0elsuppb
StepHypRef Expression
1 fvn0elsupp 7996 . . . 4 (((𝐵𝑉𝑋𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅))
21exp43 437 . . 3 (𝐵𝑉 → (𝑋𝐵 → (𝐺 Fn 𝐵 → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))))
323imp 1110 . 2 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))
4 simp3 1137 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
5 simp1 1135 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐵𝑉)
6 0ex 5231 . . . . 5 ∅ ∈ V
76a1i 11 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ∅ ∈ V)
8 elsuppfn 7987 . . . 4 ((𝐺 Fn 𝐵𝐵𝑉 ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
94, 5, 7, 8syl3anc 1370 . . 3 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
10 simpr 485 . . 3 ((𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅) → (𝐺𝑋) ≠ ∅)
119, 10syl6bi 252 . 2 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) → (𝐺𝑋) ≠ ∅))
123, 11impbid 211 1 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wcel 2106  wne 2943  Vcvv 3432  c0 4256   Fn wfn 6428  cfv 6433  (class class class)co 7275   supp csupp 7977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-supp 7978
This theorem is referenced by:  brcic  17510
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