Theorem symgval 19250

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.)

Hypotheses

Ref	Expression
symgval.1	⊢ 𝐺 = (SymGrp‘𝐴)
symgval.2	⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}

Assertion

Ref	Expression
symgval	⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐺(𝑥)

Proof of Theorem symgval

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		symgval.1	. 2 ⊢ 𝐺 = (SymGrp‘𝐴)
2		df-symg 19249	. . . . 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 6822	. . . . . 6 ⊢ (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴))
5		eqidd 2730	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ℎ = ℎ)
6		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7	5, 6, 6	f1oeq123d 6758	. . . . . . . . 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 6752	. . . . . . . . 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 2780	. . . . . . 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 2782	. . . . . 6 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = 𝐵)
14	4, 13	oveq12d 7367	. . . . 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 7384	. . . 4 ⊢ (𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) ∈ V)
18		nfv 1914	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V
19		nfcv 2891	. . . 4 ⊢ Ⅎ𝑥𝐴
20		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥(EndoFMnd‘𝐴)
21		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥 ↾s
22		nfab1 2893	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
23	12, 22	nfcxfr 2889	. . . . 5 ⊢ Ⅎ𝑥𝐵
24	20, 21, 23	nfov 7379	. . . 4 ⊢ Ⅎ𝑥((EndoFMnd‘𝐴) ↾s 𝐵)
25	3, 15, 16, 17, 18, 19, 24	fvmptdf 6936	. . 3 ⊢ (𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵))
26		ress0 17154	. . . . 5 ⊢ (∅ ↾s 𝐵) = ∅
27	26	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐵) = ∅)
28		fvprc 6814	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅)
29	28	oveq1d 7364	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
30		fvprc 6814	. . . 4 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅)
31	27, 29, 30	3eqtr4rd 2775	. . 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 2752	1 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3436  ∅c0 4284   ↦ cmpt 5173  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349   ↾_s cress 17141  EndoFMndcefmnd 18742  SymGrpcsymg 19248

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-1cn 11067  ax-addcl 11069

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-nn 12129  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-symg 19249

This theorem is referenced by:  symgbas  19251  symgressbas  19261  symgplusg  19262  symgvalstruct  19276  symgtset  19278