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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 19387 | . . . . 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 6906 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴)) | |
| 5 | eqidd 2738 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ℎ = ℎ) | |
| 6 | id 22 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 7 | 5, 6, 6 | f1oeq123d 6842 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (ℎ:𝑥–1-1-onto→𝑥 ↔ ℎ:𝐴–1-1-onto→𝐴)) | 
| 8 | 7 | abbidv 2808 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴}) | 
| 9 | f1oeq1 6836 | . . . . . . . . 9 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1-onto→𝐴 ↔ 𝑥:𝐴–1-1-onto→𝐴)) | |
| 10 | 9 | cbvabv 2812 | . . . . . . . 8 ⊢ {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} | 
| 11 | 8, 10 | eqtrdi 2793 | . . . . . . 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 2795 | . . . . . 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 1914 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V | |
| 19 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 20 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥(EndoFMnd‘𝐴) | |
| 21 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥 ↾s | |
| 22 | nfab1 2907 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} | |
| 23 | 12, 22 | nfcxfr 2903 | . . . . 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 6898 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅) | |
| 29 | 28 | oveq1d 7446 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵)) | 
| 30 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅) | |
| 31 | 27, 29, 30 | 3eqtr4rd 2788 | . . 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 2765 | 1 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∅c0 4333 ↦ cmpt 5225 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ↾s cress 17274 EndoFMndcefmnd 18881 SymGrpcsymg 19386 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-symg 19387 | 
| This theorem is referenced by: symgbas 19389 symgressbas 19399 symgplusg 19400 symgvalstruct 19414 symgvalstructOLD 19415 symgtset 19417 | 
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