Theorem symgval 19615

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⟶1-1onto𝐴}

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 19614	. . . . 5 ⊢ SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥⟶1-1onto𝑥}))
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ V → SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥⟶1-1onto𝑥})))
4		fveq2 6908	. . . . . 6 ⊢ (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴))
5		eqidd 2764	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ℎ = ℎ)
6		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
7	5, 6, 6	f1oeq123d 6840	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (ℎ:𝑥⟶1-1onto𝑥 ↔ ℎ:𝐴⟶1-1onto𝐴))
8	7	abbidv 2829	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥⟶1-1onto𝑥} = {ℎ ∣ ℎ:𝐴⟶1-1onto𝐴})
9		f1oeq1 6834	. . . . . . . . 9 ⊢ (ℎ = 𝑥 → (ℎ:𝐴⟶1-1onto𝐴 ↔ 𝑥:𝐴⟶1-1onto𝐴))
10	9	cbvabv 2834	. . . . . . . 8 ⊢ {ℎ ∣ ℎ:𝐴⟶1-1onto𝐴} = {𝑥 ∣ 𝑥:𝐴⟶1-1onto𝐴}
11	8, 10	eqtrdi 2814	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥⟶1-1onto𝑥} = {𝑥 ∣ 𝑥:𝐴⟶1-1onto𝐴})
12		symgval.2	. . . . . . 7 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴⟶1-1onto𝐴}
13	11, 12	eqtr4di 2816	. . . . . 6 ⊢ (𝑥 = 𝐴 → {ℎ ∣ ℎ:𝑥⟶1-1onto𝑥} = 𝐵)
14	4, 13	oveq12d 7461	. . . . 5 ⊢ (𝑥 = 𝐴 → ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥⟶1-1onto𝑥}) = ((EndoFMnd‘𝐴) ↾s 𝐵))
15	14	adantl 487	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑥 = 𝐴) → ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥⟶1-1onto𝑥}) = ((EndoFMnd‘𝐴) ↾s 𝐵))
16		id 23	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
17		ovexd 7478	. . . 4 ⊢ (𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) ∈ V)
18		nfv 1949	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V
19		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥𝐴
20		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥(EndoFMnd‘𝐴)
21		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥 ↾s
22		nfab1 2928	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥:𝐴⟶1-1onto𝐴}
23	12, 22	nfcxfr 2924	. . . . 5 ⊢ Ⅎ𝑥𝐵
24	20, 21, 23	nfov 7473	. . . 4 ⊢ Ⅎ𝑥((EndoFMnd‘𝐴) ↾s 𝐵)
25	3, 15, 16, 17, 18, 19, 24	fvmptdf 7023	. . 3 ⊢ (𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵))
26		ress0 17450	. . . . 5 ⊢ (∅ ↾s 𝐵) = ∅
27	26	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → (∅ ↾s 𝐵) = ∅)
28		fvprc 6900	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅)
29	28	oveq1d 7458	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
30		fvprc 6900	. . . 4 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅)
31	27, 29, 30	3eqtr4rd 2809	. . 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 2786	1 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  {cab 2741  Vcvv 3454  ∅c0 4281   ↦ cmpt 5189  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443   ↾_s cress 17437  EndoFMndcefmnd 19093  SymGrpcsymg 19613

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-1cn 11285  ax-addcl 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-nn 12363  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-symg 19614

This theorem is used by:  symgbas  19616  symgressbas  19626  symgplusg  19627  symgvalstruct  19641  symgtset  19643