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Theorem fvn0elsuppb 8168
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 8167 . . . 4 (((𝐵𝑉𝑋𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅))
21exp43 437 . . 3 (𝐵𝑉 → (𝑋𝐵 → (𝐺 Fn 𝐵 → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))))
323imp 1111 . 2 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))
4 simp3 1138 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
5 simp1 1136 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐵𝑉)
6 0ex 5307 . . . . 5 ∅ ∈ V
76a1i 11 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ∅ ∈ V)
8 elsuppfn 8158 . . . 4 ((𝐺 Fn 𝐵𝐵𝑉 ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
94, 5, 7, 8syl3anc 1371 . . 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 1087  wcel 2106  wne 2940  Vcvv 3474  c0 4322   Fn wfn 6538  cfv 6543  (class class class)co 7411   supp csupp 8148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-supp 8149
This theorem is referenced by:  brcic  17747
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