Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5809 | . . . . . . 7 ⊢ (𝐺:𝑋–1-1→𝑌 ↔ (𝐺:𝑋⟶𝑌 ∧ Fun ◡𝐺)) | |
4 | 3 | simprbi 479 | . . . . . 6 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun ◡𝐺) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun ◡𝐺) |
6 | cofunex2g 7024 | . . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ Fun ◡𝐺) → (𝐹 ∘ 𝐺) ∈ V) | |
7 | 1, 5, 6 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ V) |
8 | fsuppco.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
9 | suppimacnv 7193 | . . . 4 ⊢ (((𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) | |
10 | 7, 8, 9 | syl2anc 691 | . . 3 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍}))) |
11 | suppimacnv 7193 | . . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | |
12 | 1, 8, 11 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) |
13 | fsuppco.f | . . . . . 6 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
14 | 13 | fsuppimpd 8165 | . . . . 5 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
15 | 12, 14 | eqeltrrd 2689 | . . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑍})) ∈ Fin) |
16 | 15, 2 | fsuppcolem 8189 | . . 3 ⊢ (𝜑 → (◡(𝐹 ∘ 𝐺) “ (V ∖ {𝑍})) ∈ Fin) |
17 | 10, 16 | eqeltrd 2688 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin) |
18 | fsuppimp 8164 | . . . . . 6 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
19 | 18 | simpld 474 | . . . . 5 ⊢ (𝐹 finSupp 𝑍 → Fun 𝐹) |
20 | 13, 19 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
21 | f1fun 6016 | . . . . 5 ⊢ (𝐺:𝑋–1-1→𝑌 → Fun 𝐺) | |
22 | 2, 21 | syl 17 | . . . 4 ⊢ (𝜑 → Fun 𝐺) |
23 | funco 5842 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
24 | 20, 22, 23 | syl2anc 691 | . . 3 ⊢ (𝜑 → Fun (𝐹 ∘ 𝐺)) |
25 | funisfsupp 8163 | . . 3 ⊢ ((Fun (𝐹 ∘ 𝐺) ∧ (𝐹 ∘ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) | |
26 | 24, 7, 8, 25 | syl3anc 1318 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) finSupp 𝑍 ↔ ((𝐹 ∘ 𝐺) supp 𝑍) ∈ Fin)) |
27 | 17, 26 | mpbird 246 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 {csn 4125 class class class wbr 4583 ◡ccnv 5037 “ cima 5041 ∘ ccom 5042 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 (class class class)co 6549 supp csupp 7182 Fincfn 7841 finSupp cfsupp 8158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-supp 7183 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-fin 7845 df-fsupp 8159 |
This theorem is referenced by: mapfienlem1 8193 mapfienlem2 8194 coe1sfi 19404 |
Copyright terms: Public domain | W3C validator |