Theorem supp0cosupp0 7855

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 7853	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
2		imaeq2 5892	. . . 4 ⊢ ((𝐹 supp 𝑍) = ∅ → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ ∅))
3		ima0 5912	. . . 4 ⊢ (◡𝐺 “ ∅) = ∅
4	2, 3	eqtrdi 2849	. . 3 ⊢ ((𝐹 supp 𝑍) = ∅ → (◡𝐺 “ (𝐹 supp 𝑍)) = ∅)
5	1, 4	sylan9eq 2853	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (𝐹 supp 𝑍) = ∅) → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)
6	5	ex 416	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∅c0 4243  ^◡ccnv 5518   “ cima 5522   ∘ ccom 5523  (class class class)co 7135   supp csupp 7813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-supp 7814

This theorem is referenced by:  gsumval3lem2  19019