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Mirrors > Home > MPE Home > Th. List > frnfsuppbi | Structured version Visualization version GIF version |
Description: Two ways of saying that a function with known codomain is finitely supported. (Contributed by AV, 8-Jul-2019.) |
Ref | Expression |
---|---|
frnfsuppbi | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 finSupp 𝑍 ↔ (◡𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffun 6510 | . . . . 5 ⊢ (𝐹:𝐼⟶𝑆 → Fun 𝐹) | |
2 | 1 | adantl 482 | . . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → Fun 𝐹) |
3 | fex 6980 | . . . . . . 7 ⊢ ((𝐹:𝐼⟶𝑆 ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 3 | expcom 414 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V)) |
5 | 4 | adantr 481 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V)) |
6 | 5 | imp 407 | . . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝐹 ∈ V) |
7 | simplr 765 | . . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝑍 ∈ 𝑊) | |
8 | funisfsupp 8826 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) | |
9 | 2, 6, 7, 8 | syl3anc 1363 | . . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) |
10 | frnsuppeq 7831 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍})))) | |
11 | 10 | imp 407 | . . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))) |
12 | 11 | eleq1d 2894 | . . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → ((𝐹 supp 𝑍) ∈ Fin ↔ (◡𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin)) |
13 | 9, 12 | bitrd 280 | . 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 finSupp 𝑍 ↔ (◡𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin)) |
14 | 13 | ex 413 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 finSupp 𝑍 ↔ (◡𝐹 “ (𝑆 ∖ {𝑍})) ∈ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 {csn 4557 class class class wbr 5057 ◡ccnv 5547 “ cima 5551 Fun wfun 6342 ⟶wf 6344 (class class class)co 7145 supp csupp 7819 Fincfn 8497 finSupp cfsupp 8821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-supp 7820 df-fsupp 8822 |
This theorem is referenced by: frnnn0fsupp 11942 |
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