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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 19299 | . . . . 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 6834 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴)) | |
| 5 | eqidd 2737 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ℎ = ℎ) | |
| 6 | id 22 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 7 | 5, 6, 6 | f1oeq123d 6768 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (ℎ:𝑥–1-1-onto→𝑥 ↔ ℎ:𝐴–1-1-onto→𝐴)) |
| 8 | 7 | abbidv 2802 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴}) |
| 9 | f1oeq1 6762 | . . . . . . . . 9 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1-onto→𝐴 ↔ 𝑥:𝐴–1-1-onto→𝐴)) | |
| 10 | 9 | cbvabv 2806 | . . . . . . . 8 ⊢ {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} |
| 11 | 8, 10 | eqtrdi 2787 | . . . . . . 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 2789 | . . . . . 6 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = 𝐵) |
| 14 | 4, 13 | oveq12d 7376 | . . . . 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 7393 | . . . 4 ⊢ (𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) ∈ V) | |
| 18 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V | |
| 19 | nfcv 2898 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 20 | nfcv 2898 | . . . . 5 ⊢ Ⅎ𝑥(EndoFMnd‘𝐴) | |
| 21 | nfcv 2898 | . . . . 5 ⊢ Ⅎ𝑥 ↾s | |
| 22 | nfab1 2900 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} | |
| 23 | 12, 22 | nfcxfr 2896 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
| 24 | 20, 21, 23 | nfov 7388 | . . . 4 ⊢ Ⅎ𝑥((EndoFMnd‘𝐴) ↾s 𝐵) |
| 25 | 3, 15, 16, 17, 18, 19, 24 | fvmptdf 6947 | . . 3 ⊢ (𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵)) |
| 26 | ress0 17170 | . . . . 5 ⊢ (∅ ↾s 𝐵) = ∅ | |
| 27 | 26 | a1i 11 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐵) = ∅) |
| 28 | fvprc 6826 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅) | |
| 29 | 28 | oveq1d 7373 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵)) |
| 30 | fvprc 6826 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅) | |
| 31 | 27, 29, 30 | 3eqtr4rd 2782 | . . 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 2759 | 1 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2113 {cab 2714 Vcvv 3440 ∅c0 4285 ↦ cmpt 5179 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 ↾s cress 17157 EndoFMndcefmnd 18793 SymGrpcsymg 19298 |
| 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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-1cn 11084 ax-addcl 11086 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-nn 12146 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-symg 19299 |
| This theorem is referenced by: symgbas 19301 symgressbas 19311 symgplusg 19312 symgvalstruct 19326 symgtset 19328 |
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