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Mirrors > Home > MPE Home > Th. List > supp0cosupp0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
supp0cosupp0 | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppco 7870 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))) | |
2 | imaeq2 5925 | . . . 4 ⊢ ((𝐹 supp 𝑍) = ∅ → (◡𝐺 “ (𝐹 supp 𝑍)) = (◡𝐺 “ ∅)) | |
3 | ima0 5945 | . . . 4 ⊢ (◡𝐺 “ ∅) = ∅ | |
4 | 2, 3 | syl6eq 2872 | . . 3 ⊢ ((𝐹 supp 𝑍) = ∅ → (◡𝐺 “ (𝐹 supp 𝑍)) = ∅) |
5 | 1, 4 | sylan9eq 2876 | . 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ (𝐹 supp 𝑍) = ∅) → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅) |
6 | 5 | ex 415 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∅c0 4291 ◡ccnv 5554 “ cima 5558 ∘ ccom 5559 (class class class)co 7156 supp csupp 7830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-supp 7831 |
This theorem is referenced by: gsumval3lem2 19026 |
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