Theorem supp0cosupp0 8206

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	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))

Proof of Theorem supp0cosupp0

Step	Hyp	Ref	Expression
1		suppco 8204	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
2		imaeq2 6060	. . . 4 ⊢ ((𝐹 supp 𝑍) = ∅ → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ ∅))
3		ima0 6081	. . . 4 ⊢ (◡𝐺 “ ∅) = ∅
4	2, 3	eqtrdi 2816	. . 3 ⊢ ((𝐹 supp 𝑍) = ∅ → (◡𝐺 “ (𝐹 supp 𝑍)) = ∅)
5	1, 4	sylan9eq 2820	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (𝐹 supp 𝑍) = ∅) → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
6	5	ex 418	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∅c0 4286  ^◡ccnv 5662   “ cima 5666   ∘ ccom 5667  (class class class)co 7416   supp csupp 8158

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-supp 8159

This theorem is used by:  gsumval3lem2  20000