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Mirrors > Home > MPE Home > Th. List > ressuppfi | Structured version Visualization version GIF version |
Description: If the support of the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finite, the support of the function itself is finite. (Contributed by AV, 22-Apr-2019.) |
Ref | Expression |
---|---|
ressuppfi.b | ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) |
ressuppfi.f | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
ressuppfi.g | ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵)) |
ressuppfi.s | ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) |
ressuppfi.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
ressuppfi | ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressuppfi.g | . . . . . 6 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ 𝐵)) | |
2 | 1 | eqcomd 2829 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = 𝐺) |
3 | 2 | oveq1d 7173 | . . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) = (𝐺 supp 𝑍)) |
4 | ressuppfi.s | . . . 4 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) | |
5 | 3, 4 | eqeltrd 2915 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin) |
6 | ressuppfi.b | . . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) | |
7 | unfi 8787 | . . 3 ⊢ ((((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹 ∖ 𝐵) ∈ Fin) → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin) | |
8 | 5, 6, 7 | syl2anc 586 | . 2 ⊢ (𝜑 → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin) |
9 | ressuppfi.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
10 | ressuppfi.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
11 | ressuppssdif 7853 | . . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵))) | |
12 | 9, 10, 11 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵))) |
13 | 8, 12 | ssfid 8743 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ∪ cun 3936 ⊆ wss 3938 dom cdm 5557 ↾ cres 5559 (class class class)co 7158 supp csupp 7832 Fincfn 8511 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-er 8291 df-en 8512 df-fin 8515 |
This theorem is referenced by: resfsupp 8862 |
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