Theorem symgval 18499

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 18498	. . . . 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 6672	. . . . . 6 ⊢ (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴))
5		eqidd 2824	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ℎ = ℎ)
6		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7	5, 6, 6	f1oeq123d 6612	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (ℎ:𝑥–1-1-onto→𝑥 ↔ ℎ:𝐴–1-1-onto→𝐴))
8	7	abbidv 2887	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴})
9		f1oeq1 6606	. . . . . . . . 9 ⊢ (ℎ = 𝑥 → (ℎ:𝐴–1-1-onto→𝐴 ↔ 𝑥:𝐴–1-1-onto→𝐴))
10	9	cbvabv 2891	. . . . . . . 8 ⊢ {ℎ ∣ ℎ:𝐴–1-1-onto→𝐴} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
11	8, 10	syl6eq 2874	. . . . . . 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	syl6eqr 2876	. . . . . 6 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = 𝐵)
14	4, 13	oveq12d 7176	. . . . 5 ⊢ (𝑥 = 𝐴 → ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}) = ((EndoFMnd‘𝐴) ↾s 𝐵))
15	14	adantl 484	. . . 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 7193	. . . 4 ⊢ (𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) ∈ V)
18		nfv 1915	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V
19		nfcv 2979	. . . 4 ⊢ Ⅎ𝑥𝐴
20		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑥(EndoFMnd‘𝐴)
21		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑥 ↾s
22		nfab1 2981	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
23	12, 22	nfcxfr 2977	. . . . 5 ⊢ Ⅎ𝑥𝐵
24	20, 21, 23	nfov 7188	. . . 4 ⊢ Ⅎ𝑥((EndoFMnd‘𝐴) ↾s 𝐵)
25	3, 15, 16, 17, 18, 19, 24	fvmptdf 6776	. . 3 ⊢ (𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵))
26		ress0 16560	. . . . 5 ⊢ (∅ ↾s 𝐵) = ∅
27	26	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐵) = ∅)
28		fvprc 6665	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅)
29	28	oveq1d 7173	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
30		fvprc 6665	. . . 4 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅)
31	27, 29, 30	3eqtr4rd 2869	. . 3 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵))
32	25, 31	pm2.61i 184	. 2 ⊢ (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵)
33	1, 32	eqtri 2846	1 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  {cab 2801  Vcvv 3496  ∅c0 4293   ↦ cmpt 5148  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158   ↾_s cress 16486  EndoFMndcefmnd 18035  SymGrpcsymg 18497

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-slot 16489  df-base 16491  df-ress 16493  df-symg 18498

This theorem is referenced by:  symgbas  18501  symgressbas  18512  symgplusg  18513  symgvalstruct  18527  symgtset  18529