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Theorem supp0cosupp0 7973
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 7971 . . 3 ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
2 imaeq2 5942 . . . 4 ((𝐹 supp 𝑍) = ∅ → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ ∅))
3 ima0 5962 . . . 4 (𝐺 “ ∅) = ∅
42, 3eqtrdi 2796 . . 3 ((𝐹 supp 𝑍) = ∅ → (𝐺 “ (𝐹 supp 𝑍)) = ∅)
51, 4sylan9eq 2800 . 2 (((𝐹𝑉𝐺𝑊) ∧ (𝐹 supp 𝑍) = ∅) → ((𝐹𝐺) supp 𝑍) = ∅)
65ex 416 1 ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2112  c0 4253  ccnv 5567  cima 5571  ccom 5572  (class class class)co 7234   supp csupp 7926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-iota 6358  df-fun 6402  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239  df-supp 7927
This theorem is referenced by:  gsumval3lem2  19323
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