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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 2827 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = 𝐺) |
3 | 2 | oveq1d 7165 | . . . 4 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) = (𝐺 supp 𝑍)) |
4 | ressuppfi.s | . . . 4 ⊢ (𝜑 → (𝐺 supp 𝑍) ∈ Fin) | |
5 | 3, 4 | eqeltrd 2913 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) supp 𝑍) ∈ Fin) |
6 | ressuppfi.b | . . 3 ⊢ (𝜑 → (dom 𝐹 ∖ 𝐵) ∈ Fin) | |
7 | unfi 8779 | . . 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 7845 | . . 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 8735 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ∪ cun 3933 ⊆ wss 3935 dom cdm 5549 ↾ cres 5551 (class class class)co 7150 supp csupp 7824 Fincfn 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-oadd 8100 df-er 8283 df-en 8504 df-fin 8507 |
This theorem is referenced by: resfsupp 8854 |
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