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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 19402 | . . . . 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 6907 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴)) | |
5 | eqidd 2736 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ℎ = ℎ) | |
6 | id 22 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
7 | 5, 6, 6 | f1oeq123d 6843 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (ℎ:𝑥–1-1-onto→𝑥 ↔ ℎ:𝐴–1-1-onto→𝐴)) |
8 | 7 | abbidv 2806 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴}) |
9 | f1oeq1 6837 | . . . . . . . . 9 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1-onto→𝐴 ↔ 𝑥:𝐴–1-1-onto→𝐴)) | |
10 | 9 | cbvabv 2810 | . . . . . . . 8 ⊢ {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} |
11 | 8, 10 | eqtrdi 2791 | . . . . . . 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 2793 | . . . . . 6 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = 𝐵) |
14 | 4, 13 | oveq12d 7449 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}) = ((EndoFMnd‘𝐴) ↾s 𝐵)) |
15 | 14 | adantl 481 | . . . 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 7466 | . . . 4 ⊢ (𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) ∈ V) | |
18 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V | |
19 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
20 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥(EndoFMnd‘𝐴) | |
21 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥 ↾s | |
22 | nfab1 2905 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} | |
23 | 12, 22 | nfcxfr 2901 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
24 | 20, 21, 23 | nfov 7461 | . . . 4 ⊢ Ⅎ𝑥((EndoFMnd‘𝐴) ↾s 𝐵) |
25 | 3, 15, 16, 17, 18, 19, 24 | fvmptdf 7022 | . . 3 ⊢ (𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵)) |
26 | ress0 17289 | . . . . 5 ⊢ (∅ ↾s 𝐵) = ∅ | |
27 | 26 | a1i 11 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐵) = ∅) |
28 | fvprc 6899 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅) | |
29 | 28 | oveq1d 7446 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵)) |
30 | fvprc 6899 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅) | |
31 | 27, 29, 30 | 3eqtr4rd 2786 | . . 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 2763 | 1 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 {cab 2712 Vcvv 3478 ∅c0 4339 ↦ cmpt 5231 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ↾s cress 17274 EndoFMndcefmnd 18894 SymGrpcsymg 19401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-symg 19402 |
This theorem is referenced by: symgbas 19404 symgressbas 19414 symgplusg 19415 symgvalstruct 19429 symgvalstructOLD 19430 symgtset 19432 |
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