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Theorem fvn0elsuppb 7705
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 7704 . . . 4 (((𝐵𝑉𝑋𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅))
21exp43 437 . . 3 (𝐵𝑉 → (𝑋𝐵 → (𝐺 Fn 𝐵 → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))))
323imp 1104 . 2 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))
4 simp3 1131 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
5 simp1 1129 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐵𝑉)
6 0ex 5109 . . . . 5 ∅ ∈ V
76a1i 11 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ∅ ∈ V)
8 elsuppfn 7696 . . . 4 ((𝐺 Fn 𝐵𝐵𝑉 ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
94, 5, 7, 8syl3anc 1364 . . 3 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
10 simpr 485 . . 3 ((𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅) → (𝐺𝑋) ≠ ∅)
119, 10syl6bi 254 . 2 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) → (𝐺𝑋) ≠ ∅))
123, 11impbid 213 1 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080  wcel 2083  wne 2986  Vcvv 3440  c0 4217   Fn wfn 6227  cfv 6232  (class class class)co 7023   supp csupp 7688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-supp 7689
This theorem is referenced by:  brcic  16901
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