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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 8873. (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 5071 | . . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂)) | |
3 | 2 | cbvrabv 3493 | . . . 4 ⊢ {ℎ ∈ (𝑆 ↑m 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂} |
4 | 1, 3 | eqtr4i 2849 | . . 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 6654 | . . . 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 3514 | . . . 4 ⊢ (𝑅 ∈ 𝑈 → 𝑅 ∈ V) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
13 | fcobij.3 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
14 | elex 3514 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
16 | fcobij.4 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑊) | |
17 | elex 3514 | . . . 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 8873 | . 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌) |
21 | ssrab2 4058 | . . . . . . 7 ⊢ {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂} ⊆ (𝑆 ↑m 𝑅) | |
22 | 1, 21 | eqsstri 4003 | . . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑m 𝑅) |
23 | 22 | sseli 3965 | . . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑m 𝑅)) |
24 | coass 6120 | . . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) | |
25 | f1of 6617 | . . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇) | |
26 | 9, 25 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇) |
27 | elmapi 8430 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑅) → 𝑓:𝑅⟶𝑆) | |
28 | fco 6533 | . . . . . . . 8 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇) | |
29 | 26, 27, 28 | syl2an 597 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇) |
30 | fcoi1 6554 | . . . . . . 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 2872 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓)) |
33 | 23, 32 | sylan2 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓)) |
34 | 33 | mpteq2dva 5163 | . . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓))) |
35 | f1oeq1 6606 | . . 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 234 | 1 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 class class class wbr 5068 ↦ cmpt 5148 I cid 5461 ↾ cres 5559 ∘ ccom 5561 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 finSupp cfsupp 8835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-1o 8104 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-fin 8515 df-fsupp 8836 |
This theorem is referenced by: eulerpartgbij 31632 |
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