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Theorem fvn0elsuppb 7888
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 7887 . . . 4 (((𝐵𝑉𝑋𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅))
21exp43 440 . . 3 (𝐵𝑉 → (𝑋𝐵 → (𝐺 Fn 𝐵 → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))))
323imp 1112 . 2 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ → 𝑋 ∈ (𝐺 supp ∅)))
4 simp3 1139 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
5 simp1 1137 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → 𝐵𝑉)
6 0ex 5185 . . . . 5 ∅ ∈ V
76a1i 11 . . . 4 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ∅ ∈ V)
8 elsuppfn 7878 . . . 4 ((𝐺 Fn 𝐵𝐵𝑉 ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
94, 5, 7, 8syl3anc 1372 . . 3 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
10 simpr 488 . . 3 ((𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅) → (𝐺𝑋) ≠ ∅)
119, 10syl6bi 256 . 2 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → (𝑋 ∈ (𝐺 supp ∅) → (𝐺𝑋) ≠ ∅))
123, 11impbid 215 1 ((𝐵𝑉𝑋𝐵𝐺 Fn 𝐵) → ((𝐺𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088  wcel 2114  wne 2935  Vcvv 3400  c0 4221   Fn wfn 6344  cfv 6349  (class class class)co 7182   supp csupp 7868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7185  df-oprab 7186  df-mpo 7187  df-supp 7869
This theorem is referenced by:  brcic  17185
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