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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 2739 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = 𝐺) |
3 | 2 | oveq1d 7424 | . . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) = (𝐺 supp 𝑍)) |
4 | ressuppfi.s | . . . 4 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) | |
5 | 3, 4 | eqeltrd 2834 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin) |
6 | ressuppfi.b | . . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) | |
7 | unfi 9172 | . . 3 ⊢ ((((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹 ∖ 𝐵) ∈ Fin) → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin) | |
8 | 5, 6, 7 | syl2anc 585 | . 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 585 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵))) |
13 | 8, 12 | ssfid 9267 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 ∪ cun 3947 ⊆ wss 3949 dom cdm 5677 ↾ cres 5679 (class class class)co 7409 supp csupp 8146 Fincfn 8939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-supp 8147 df-1o 8466 df-en 8940 df-fin 8943 |
This theorem is referenced by: resfsupp 9391 |
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