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Mirrors > Home > MPE Home > Th. List > fidmfisupp | Structured version Visualization version GIF version |
Description: A function with a finite domain is finitely supported. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
fidmfisupp.1 | ⊢ (𝜑 → 𝐹:𝐷⟶𝑅) |
fidmfisupp.2 | ⊢ (𝜑 → 𝐷 ∈ Fin) |
fidmfisupp.3 | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
fidmfisupp | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fidmfisupp.1 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶𝑅) | |
2 | fidmfisupp.2 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ Fin) | |
3 | 1, 2 | fexd 7178 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
4 | fidmfisupp.3 | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
5 | suppimacnv 8106 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | |
6 | 3, 4, 5 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) |
7 | 2, 1 | fisuppfi 9317 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) |
8 | 6, 7 | eqeltrd 2834 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
9 | 1 | ffund 6673 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
10 | funisfsupp 9314 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) | |
11 | 9, 3, 4, 10 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) |
12 | 8, 11 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∖ cdif 3908 {csn 4587 class class class wbr 5106 ◡ccnv 5633 “ cima 5637 Fun wfun 6491 ⟶wf 6493 (class class class)co 7358 supp csupp 8093 Fincfn 8886 finSupp cfsupp 9308 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-supp 8094 df-1o 8413 df-en 8887 df-fin 8890 df-fsupp 9309 |
This theorem is referenced by: mptiffisupp 31654 rrxtopnfi 44614 |
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