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Mirrors > Home > MPE Home > Th. List > symgval | Structured version Visualization version GIF version |
Description: The value of the symmetric group function at 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 28-Mar-2024.) |
Ref | Expression |
---|---|
symgval.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symgval.2 | ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} |
Ref | Expression |
---|---|
symgval | ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgval.1 | . 2 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | df-symg 19361 | . . . . 5 ⊢ SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥})) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ V → SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}))) |
4 | fveq2 6893 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴)) | |
5 | eqidd 2727 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ℎ = ℎ) | |
6 | id 22 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
7 | 5, 6, 6 | f1oeq123d 6829 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (ℎ:𝑥–1-1-onto→𝑥 ↔ ℎ:𝐴–1-1-onto→𝐴)) |
8 | 7 | abbidv 2795 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴}) |
9 | f1oeq1 6823 | . . . . . . . . 9 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1-onto→𝐴 ↔ 𝑥:𝐴–1-1-onto→𝐴)) | |
10 | 9 | cbvabv 2799 | . . . . . . . 8 ⊢ {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} |
11 | 8, 10 | eqtrdi 2782 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}) |
12 | symgval.2 | . . . . . . 7 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} | |
13 | 11, 12 | eqtr4di 2784 | . . . . . 6 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = 𝐵) |
14 | 4, 13 | oveq12d 7434 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}) = ((EndoFMnd‘𝐴) ↾s 𝐵)) |
15 | 14 | adantl 480 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑥 = 𝐴) → ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}) = ((EndoFMnd‘𝐴) ↾s 𝐵)) |
16 | id 22 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
17 | ovexd 7451 | . . . 4 ⊢ (𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) ∈ V) | |
18 | nfv 1910 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V | |
19 | nfcv 2892 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
20 | nfcv 2892 | . . . . 5 ⊢ Ⅎ𝑥(EndoFMnd‘𝐴) | |
21 | nfcv 2892 | . . . . 5 ⊢ Ⅎ𝑥 ↾s | |
22 | nfab1 2894 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} | |
23 | 12, 22 | nfcxfr 2890 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
24 | 20, 21, 23 | nfov 7446 | . . . 4 ⊢ Ⅎ𝑥((EndoFMnd‘𝐴) ↾s 𝐵) |
25 | 3, 15, 16, 17, 18, 19, 24 | fvmptdf 7007 | . . 3 ⊢ (𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵)) |
26 | ress0 17252 | . . . . 5 ⊢ (∅ ↾s 𝐵) = ∅ | |
27 | 26 | a1i 11 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐵) = ∅) |
28 | fvprc 6885 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅) | |
29 | 28 | oveq1d 7431 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵)) |
30 | fvprc 6885 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅) | |
31 | 27, 29, 30 | 3eqtr4rd 2777 | . . 3 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵)) |
32 | 25, 31 | pm2.61i 182 | . 2 ⊢ (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵) |
33 | 1, 32 | eqtri 2754 | 1 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 {cab 2703 Vcvv 3462 ∅c0 4322 ↦ cmpt 5228 –1-1-onto→wf1o 6545 ‘cfv 6546 (class class class)co 7416 ↾s cress 17237 EndoFMndcefmnd 18853 SymGrpcsymg 19360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-1cn 11207 ax-addcl 11209 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-nn 12259 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-symg 19361 |
This theorem is referenced by: symgbas 19364 symgressbas 19375 symgplusg 19376 symgvalstruct 19390 symgvalstructOLD 19391 symgtset 19393 |
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