Theorem symgval 19308

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 19307	. . . . 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 6861	. . . . . 6 ⊢ (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴))
5		eqidd 2731	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ℎ = ℎ)
6		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7	5, 6, 6	f1oeq123d 6797	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (ℎ:𝑥–1-1-onto→𝑥 ↔ ℎ:𝐴–1-1-onto→𝐴))
8	7	abbidv 2796	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴})
9		f1oeq1 6791	. . . . . . . . 9 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1-onto→𝐴 ↔ 𝑥:𝐴–1-1-onto→𝐴))
10	9	cbvabv 2800	. . . . . . . 8 ⊢ {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
11	8, 10	eqtrdi 2781	. . . . . . 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 2783	. . . . . 6 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = 𝐵)
14	4, 13	oveq12d 7408	. . . . 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 7425	. . . 4 ⊢ (𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) ∈ V)
18		nfv 1914	. . . 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 7420	. . . 4 ⊢ Ⅎ𝑥((EndoFMnd‘𝐴) ↾s 𝐵)
25	3, 15, 16, 17, 18, 19, 24	fvmptdf 6977	. . 3 ⊢ (𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵))
26		ress0 17220	. . . . 5 ⊢ (∅ ↾s 𝐵) = ∅
27	26	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐵) = ∅)
28		fvprc 6853	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅)
29	28	oveq1d 7405	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
30		fvprc 6853	. . . 4 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅)
31	27, 29, 30	3eqtr4rd 2776	. . 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 2753	1 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  {cab 2708  Vcvv 3450  ∅c0 4299   ↦ cmpt 5191  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390   ↾_s cress 17207  EndoFMndcefmnd 18802  SymGrpcsymg 19306

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-1cn 11133  ax-addcl 11135

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-nn 12194  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-symg 19307

This theorem is referenced by:  symgbas  19309  symgressbas  19319  symgplusg  19320  symgvalstruct  19334  symgtset  19336