![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fcobijfs | Structured version Visualization version GIF version |
Description: Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 9278. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
Ref | Expression |
---|---|
fcobij.1 | ⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇) |
fcobij.2 | ⊢ (𝜑 → 𝑅 ∈ 𝑈) |
fcobij.3 | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
fcobij.4 | ⊢ (𝜑 → 𝑇 ∈ 𝑊) |
fcobijfs.5 | ⊢ (𝜑 → 𝑂 ∈ 𝑆) |
fcobijfs.6 | ⊢ 𝑄 = (𝐺‘𝑂) |
fcobijfs.7 | ⊢ 𝑋 = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂} |
fcobijfs.8 | ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑄} |
Ref | Expression |
---|---|
fcobijfs | ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcobijfs.7 | . . . 4 ⊢ 𝑋 = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂} | |
2 | breq1 5107 | . . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂)) | |
3 | 2 | cbvrabv 3416 | . . . 4 ⊢ {ℎ ∈ (𝑆 ↑m 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂} |
4 | 1, 3 | eqtr4i 2769 | . . 3 ⊢ 𝑋 = {ℎ ∈ (𝑆 ↑m 𝑅) ∣ ℎ finSupp 𝑂} |
5 | fcobijfs.8 | . . 3 ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑄} | |
6 | fcobijfs.6 | . . 3 ⊢ 𝑄 = (𝐺‘𝑂) | |
7 | f1oi 6818 | . . . 4 ⊢ ( I ↾ 𝑅):𝑅–1-1-onto→𝑅 | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → ( I ↾ 𝑅):𝑅–1-1-onto→𝑅) |
9 | fcobij.1 | . . 3 ⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇) | |
10 | fcobij.2 | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑈) | |
11 | fcobij.3 | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
12 | fcobij.4 | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑊) | |
13 | fcobijfs.5 | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑆) | |
14 | 4, 5, 6, 8, 9, 10, 11, 10, 12, 13 | mapfien 9278 | . 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌) |
15 | 1 | ssrab3 4039 | . . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑m 𝑅) |
16 | 15 | sseli 3939 | . . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑m 𝑅)) |
17 | coass 6214 | . . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) | |
18 | f1of 6780 | . . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇) | |
19 | 9, 18 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇) |
20 | elmapi 8721 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑅) → 𝑓:𝑅⟶𝑆) | |
21 | fco 6688 | . . . . . . . 8 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇) | |
22 | 19, 20, 21 | syl2an 597 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇) |
23 | fcoi1 6712 | . . . . . . 7 ⊢ ((𝐺 ∘ 𝑓):𝑅⟶𝑇 → ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ 𝑓)) | |
24 | 22, 23 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ 𝑓)) |
25 | 17, 24 | eqtr3id 2792 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓)) |
26 | 16, 25 | sylan2 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓)) |
27 | 26 | mpteq2dva 5204 | . . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓))) |
28 | 27 | f1oeq1d 6775 | . 2 ⊢ (𝜑 → ((𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌 ↔ (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌)) |
29 | 14, 28 | mpbid 231 | 1 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 class class class wbr 5104 ↦ cmpt 5187 I cid 5528 ↾ cres 5633 ∘ ccom 5635 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7350 ↑m cmap 8699 finSupp cfsupp 9239 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-1o 8380 df-map 8701 df-en 8818 df-fin 8821 df-fsupp 9240 |
This theorem is referenced by: eulerpartgbij 32752 |
Copyright terms: Public domain | W3C validator |