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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 2784 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = 𝐺) |
3 | 2 | oveq1d 6939 | . . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) = (𝐺 supp 𝑍)) |
4 | ressuppfi.s | . . . 4 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) | |
5 | 3, 4 | eqeltrd 2859 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin) |
6 | ressuppfi.b | . . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) | |
7 | unfi 8517 | . . 3 ⊢ ((((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin ∧ (dom 𝐹 ∖ 𝐵) ∈ Fin) → (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)) ∈ Fin) | |
8 | 5, 6, 7 | syl2anc 579 | . 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 7599 | . . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵))) | |
12 | 9, 10, 11 | syl2anc 579 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵))) |
13 | 8, 12 | ssfid 8473 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∖ cdif 3789 ∪ cun 3790 ⊆ wss 3792 dom cdm 5357 ↾ cres 5359 (class class class)co 6924 supp csupp 7578 Fincfn 8243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-oadd 7849 df-er 8028 df-en 8244 df-fin 8247 |
This theorem is referenced by: resfsupp 8592 |
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