Theorem symgval 19362

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 19361	. . . . 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 6893	. . . . . 6 ⊢ (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴))
5		eqidd 2727	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ℎ = ℎ)
6		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7	5, 6, 6	f1oeq123d 6829	. . . . . . . . 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 6823	. . . . . . . . 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 2782	. . . . . . 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 2784	. . . . . 6 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} = 𝐵)
14	4, 13	oveq12d 7434	. . . . 5 ⊢ (𝑥 = 𝐴 → ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}) = ((EndoFMnd‘𝐴) ↾s 𝐵))
15	14	adantl 480	. . . 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 7451	. . . 4 ⊢ (𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) ∈ V)
18		nfv 1910	. . . 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 7446	. . . 4 ⊢ Ⅎ𝑥((EndoFMnd‘𝐴) ↾s 𝐵)
25	3, 15, 16, 17, 18, 19, 24	fvmptdf 7007	. . 3 ⊢ (𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵))
26		ress0 17252	. . . . 5 ⊢ (∅ ↾s 𝐵) = ∅
27	26	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐵) = ∅)
28		fvprc 6885	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅)
29	28	oveq1d 7431	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
30		fvprc 6885	. . . 4 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅)
31	27, 29, 30	3eqtr4rd 2777	. . 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 2754	1 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)

Syntax hints:  ¬ wn 3   = wceq 1534   ∈ wcel 2099  {cab 2703  Vcvv 3462  ∅c0 4322   ↦ cmpt 5228  –1-1-onto→wf1o 6545  ‘cfv 6546  (class class class)co 7416   ↾_s cress 17237  EndoFMndcefmnd 18853  SymGrpcsymg 19360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-1cn 11207  ax-addcl 11209

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-nn 12259  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-symg 19361

This theorem is referenced by:  symgbas  19364  symgressbas  19375  symgplusg  19376  symgvalstruct  19390  symgvalstructOLD  19391  symgtset  19393