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Theorem supp0cosupp0 8024
Description: The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
Assertion
Ref Expression
supp0cosupp0 ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))

Proof of Theorem supp0cosupp0
StepHypRef Expression
1 suppco 8022 . . 3 ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
2 imaeq2 5965 . . . 4 ((𝐹 supp 𝑍) = ∅ → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ ∅))
3 ima0 5985 . . . 4 (𝐺 “ ∅) = ∅
42, 3eqtrdi 2794 . . 3 ((𝐹 supp 𝑍) = ∅ → (𝐺 “ (𝐹 supp 𝑍)) = ∅)
51, 4sylan9eq 2798 . 2 (((𝐹𝑉𝐺𝑊) ∧ (𝐹 supp 𝑍) = ∅) → ((𝐹𝐺) supp 𝑍) = ∅)
65ex 413 1 ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  c0 4256  ccnv 5588  cima 5592  ccom 5593  (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-sep 5223  ax-nul 5230  ax-pow 5288  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-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  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-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-supp 7978
This theorem is referenced by:  gsumval3lem2  19507
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