Theorem supp0cosupp0 8241

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 8239	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
2		imaeq2 6081	. . . 4 ⊢ ((𝐹 supp 𝑍) = ∅ → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ ∅))
3		ima0 6102	. . . 4 ⊢ (◡𝐺 “ ∅) = ∅
4	2, 3	eqtrdi 2793	. . 3 ⊢ ((𝐹 supp 𝑍) = ∅ → (◡𝐺 “ (𝐹 supp 𝑍)) = ∅)
5	1, 4	sylan9eq 2797	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (𝐹 supp 𝑍) = ∅) → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
6	5	ex 412	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∅c0 4342  ^◡ccnv 5692   “ cima 5696   ∘ ccom 5697  (class class class)co 7438   supp csupp 8193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-supp 8194

This theorem is referenced by:  gsumval3lem2  19948