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Mirrors > Home > MPE Home > Th. List > fsuppco | Structured version Visualization version GIF version |
Description: The composition of a 1-1 function with a finitely supported function is finitely supported. (Contributed by AV, 28-May-2019.) |
Ref | Expression |
---|---|
fsuppco.f | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
fsuppco.g | ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) |
fsuppco.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
fsuppco.v | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
Ref | Expression |
---|---|
fsuppco | ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppco.v | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
2 | fsuppco.g | . . . . . 6 ⊢ (𝜑 → 𝐺:𝑋–1-1→𝑌) | |
3 | df-f1 6128 | . . . . . . 7 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺)) | |
4 | 3 | simprbi 492 | . . . . . 6 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun ◡𝐺) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun ◡𝐺) |
6 | cofunex2g 7393 | . . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ Fun ◡𝐺) → (𝐹 ∘ 𝐺) ∈ V) | |
7 | 1, 5, 6 | syl2anc 581 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
8 | fsuppco.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
9 | suppimacnv 7570 | . . . 4 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) | |
10 | 7, 8, 9 | syl2anc 581 | . . 3 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) |
11 | suppimacnv 7570 | . . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | |
12 | 1, 8, 11 | syl2anc 581 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) |
13 | fsuppco.f | . . . . . 6 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
14 | 13 | fsuppimpd 8551 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
15 | 12, 14 | eqeltrrd 2907 | . . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) |
16 | 15, 2 | fsuppcolem 8575 | . . 3 ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
17 | 10, 16 | eqeltrd 2906 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin) |
18 | fsuppimp 8550 | . . . . . 6 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
19 | 18 | simpld 490 | . . . . 5 ⊢ (𝐹 finSupp 𝑍 → Fun 𝐹) |
20 | 13, 19 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
21 | f1fun 6340 | . . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun 𝐺) | |
22 | 2, 21 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐺) |
23 | funco 6163 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
24 | 20, 22, 23 | syl2anc 581 | . . 3 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺)) |
25 | funisfsupp 8549 | . . 3 ⊢ ((Fun (𝐹 ∘ 𝐺) ∧ (𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) | |
26 | 24, 7, 8, 25 | syl3anc 1496 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) |
27 | 17, 26 | mpbird 249 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ∈ wcel 2166 Vcvv 3414 ∖ cdif 3795 {csn 4397 class class class wbr 4873 ◡ccnv 5341 “ cima 5345 ∘ ccom 5346 Fun wfun 6117 ⟶wf 6119 –1-1→wf1 6120 (class class class)co 6905 supp csupp 7559 Fincfn 8222 finSupp cfsupp 8544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-supp 7560 df-1o 7826 df-er 8009 df-en 8223 df-dom 8224 df-fin 8226 df-fsupp 8545 |
This theorem is referenced by: mapfienlem1 8579 mapfienlem2 8580 coe1sfi 19943 |
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