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Theorem supp0cosupp0 8223
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 8221 . . 3 ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
2 imaeq2 6065 . . . 4 ((𝐹 supp 𝑍) = ∅ → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ ∅))
3 ima0 6086 . . . 4 (𝐺 “ ∅) = ∅
42, 3eqtrdi 2782 . . 3 ((𝐹 supp 𝑍) = ∅ → (𝐺 “ (𝐹 supp 𝑍)) = ∅)
51, 4sylan9eq 2786 . 2 (((𝐹𝑉𝐺𝑊) ∧ (𝐹 supp 𝑍) = ∅) → ((𝐹𝐺) supp 𝑍) = ∅)
65ex 411 1 ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  c0 4325  ccnv 5681  cima 5685  ccom 5686  (class class class)co 7424   supp csupp 8174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-supp 8175
This theorem is referenced by:  gsumval3lem2  19904
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