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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 2732 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = 𝐺) |
3 | 2 | oveq1d 7420 | . . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) = (𝐺 supp 𝑍)) |
4 | ressuppfi.s | . . . 4 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) | |
5 | 3, 4 | eqeltrd 2827 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin) |
6 | ressuppfi.b | . . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) | |
7 | unfi 9174 | . . 3 ⊢ ((((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹 ∖ 𝐵) ∈ Fin) → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin) | |
8 | 5, 6, 7 | syl2anc 583 | . 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 8170 | . . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵))) | |
12 | 9, 10, 11 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵))) |
13 | 8, 12 | ssfid 9269 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∖ cdif 3940 ∪ cun 3941 ⊆ wss 3943 dom cdm 5669 ↾ cres 5671 (class class class)co 7405 supp csupp 8146 Fincfn 8941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-supp 8147 df-1o 8467 df-en 8942 df-fin 8945 |
This theorem is referenced by: resfsupp 9393 |
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