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Mathbox for Thierry Arnoux |
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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 8664. (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 4928 | . . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂)) | |
3 | 2 | cbvrabv 3405 | . . . 4 ⊢ {ℎ ∈ (𝑆 ↑𝑚 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑𝑚 𝑅) ∣ 𝑔 finSupp 𝑂} |
4 | 1, 3 | eqtr4i 2798 | . . 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 6478 | . . . 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 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈) | |
11 | elex 3426 | . . . 4 ⊢ (𝑅 ∈ 𝑈 → 𝑅 ∈ V) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
13 | fcobij.3 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
14 | elex 3426 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
16 | fcobij.4 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑊) | |
17 | elex 3426 | . . . 4 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑇 ∈ V) |
19 | fcobijfs.5 | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑆) | |
20 | 4, 5, 6, 8, 9, 12, 15, 12, 18, 19 | mapfien 8664 | . 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌) |
21 | ssrab2 3939 | . . . . . . 7 ⊢ {𝑔 ∈ (𝑆 ↑𝑚 𝑅) ∣ 𝑔 finSupp 𝑂} ⊆ (𝑆 ↑𝑚 𝑅) | |
22 | 1, 21 | eqsstri 3884 | . . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑𝑚 𝑅) |
23 | 22 | sseli 3847 | . . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) |
24 | coass 5954 | . . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) | |
25 | f1of 6441 | . . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇) | |
26 | 9, 25 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇) |
27 | elmapi 8226 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) → 𝑓:𝑅⟶𝑆) | |
28 | fco 6358 | . . . . . . . 8 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇) | |
29 | 26, 27, 28 | syl2an 587 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇) |
30 | fcoi1 6378 | . . . . . . 7 ⊢ ((𝐺 ∘ 𝑓):𝑅⟶𝑇 → ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ 𝑓)) | |
31 | 29, 30 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ 𝑓)) |
32 | 24, 31 | syl5eqr 2821 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓)) |
33 | 23, 32 | sylan2 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓)) |
34 | 33 | mpteq2dva 5018 | . . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓))) |
35 | f1oeq1 6430 | . . 3 ⊢ ((𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)) → ((𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌 ↔ (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌)) | |
36 | 34, 35 | syl 17 | . 2 ⊢ (𝜑 → ((𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌 ↔ (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌)) |
37 | 20, 36 | mpbid 224 | 1 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 {crab 3085 Vcvv 3408 class class class wbr 4925 ↦ cmpt 5004 I cid 5307 ↾ cres 5405 ∘ ccom 5407 ⟶wf 6181 –1-1-onto→wf1o 6184 ‘cfv 6185 (class class class)co 6974 ↑𝑚 cmap 8204 finSupp cfsupp 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-1o 7903 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-fin 8308 df-fsupp 8627 |
This theorem is referenced by: eulerpartgbij 31307 |
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