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 8313. (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 4656 | . . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂)) | |
3 | 2 | cbvrabv 3199 | . . . 4 ⊢ {ℎ ∈ (𝑆 ↑𝑚 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑𝑚 𝑅) ∣ 𝑔 finSupp 𝑂} |
4 | 1, 3 | eqtr4i 2647 | . . 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 6174 | . . . 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 3212 | . . . 4 ⊢ (𝑅 ∈ 𝑈 → 𝑅 ∈ V) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
13 | fcobij.3 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
14 | elex 3212 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
16 | fcobij.4 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑊) | |
17 | elex 3212 | . . . 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 8313 | . 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌) |
21 | ssrab2 3687 | . . . . . . 7 ⊢ {𝑔 ∈ (𝑆 ↑𝑚 𝑅) ∣ 𝑔 finSupp 𝑂} ⊆ (𝑆 ↑𝑚 𝑅) | |
22 | 1, 21 | eqsstri 3635 | . . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑𝑚 𝑅) |
23 | 22 | sseli 3599 | . . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) |
24 | coass 5654 | . . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) | |
25 | f1of 6137 | . . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇) | |
26 | 9, 25 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇) |
27 | elmapi 7879 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) → 𝑓:𝑅⟶𝑆) | |
28 | fco 6058 | . . . . . . . 8 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇) | |
29 | 26, 27, 28 | syl2an 494 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇) |
30 | fcoi1 6078 | . . . . . . 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 2670 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓)) |
33 | 23, 32 | sylan2 491 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓)) |
34 | 33 | mpteq2dva 4744 | . . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓))) |
35 | f1oeq1 6127 | . . 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 222 | 1 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 ↾ cres 5116 ∘ ccom 5118 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 finSupp cfsupp 8275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-1o 7560 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-fin 7959 df-fsupp 8276 |
This theorem is referenced by: eulerpartgbij 30434 |
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