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Theorem fvn0elsuppb 7514
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 7513 . . . 4 (((𝐵𝑉𝑋𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅))
21exp43 427 . . 3 (𝐵𝑉 → (𝑋𝐵 → (𝐺 Fn 𝐵 → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))))
323imp 1137 . 2 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))
4 simp3 1168 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
5 simp1 1166 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐵𝑉)
6 0ex 4950 . . . . 5 ∅ ∈ V
76a1i 11 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ∅ ∈ V)
8 elsuppfn 7505 . . . 4 ((𝐺 Fn 𝐵𝐵𝑉 ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
94, 5, 7, 8syl3anc 1490 . . 3 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
10 simpr 477 . . 3 ((𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅) → (𝐺𝑋) ≠ ∅)
119, 10syl6bi 244 . 2 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) → (𝐺𝑋) ≠ ∅))
123, 11impbid 203 1 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wcel 2155  wne 2937  Vcvv 3350  c0 4079   Fn wfn 6063  cfv 6068  (class class class)co 6842   supp csupp 7497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-supp 7498
This theorem is referenced by:  brcic  16723
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