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Theorem symgval 18488
 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 = {𝑥𝑥:𝐴1-1-onto𝐴}
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 is distinct from all other variables.
StepHypRef Expression
1 symgval.1 . 2 𝐺 = (SymGrp‘𝐴)
2 df-symg 18487 . . . . 5 SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {:𝑥1-1-onto𝑥}))
32a1i 11 . . . 4 (𝐴 ∈ V → SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {:𝑥1-1-onto𝑥})))
4 fveq2 6652 . . . . . 6 (𝑥 = 𝐴 → (EndoFMnd‘𝑥) = (EndoFMnd‘𝐴))
5 eqidd 2823 . . . . . . . . . 10 (𝑥 = 𝐴 = )
6 id 22 . . . . . . . . . 10 (𝑥 = 𝐴𝑥 = 𝐴)
75, 6, 6f1oeq123d 6592 . . . . . . . . 9 (𝑥 = 𝐴 → (:𝑥1-1-onto𝑥:𝐴1-1-onto𝐴))
87abbidv 2886 . . . . . . . 8 (𝑥 = 𝐴 → {:𝑥1-1-onto𝑥} = {:𝐴1-1-onto𝐴})
9 f1oeq1 6586 . . . . . . . . 9 ( = 𝑥 → (:𝐴1-1-onto𝐴𝑥:𝐴1-1-onto𝐴))
109cbvabv 2890 . . . . . . . 8 {:𝐴1-1-onto𝐴} = {𝑥𝑥:𝐴1-1-onto𝐴}
118, 10syl6eq 2873 . . . . . . 7 (𝑥 = 𝐴 → {:𝑥1-1-onto𝑥} = {𝑥𝑥:𝐴1-1-onto𝐴})
12 symgval.2 . . . . . . 7 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
1311, 12eqtr4di 2875 . . . . . 6 (𝑥 = 𝐴 → {:𝑥1-1-onto𝑥} = 𝐵)
144, 13oveq12d 7158 . . . . 5 (𝑥 = 𝐴 → ((EndoFMnd‘𝑥) ↾s {:𝑥1-1-onto𝑥}) = ((EndoFMnd‘𝐴) ↾s 𝐵))
1514adantl 485 . . . 4 ((𝐴 ∈ V ∧ 𝑥 = 𝐴) → ((EndoFMnd‘𝑥) ↾s {:𝑥1-1-onto𝑥}) = ((EndoFMnd‘𝐴) ↾s 𝐵))
16 id 22 . . . 4 (𝐴 ∈ V → 𝐴 ∈ V)
17 ovexd 7175 . . . 4 (𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) ∈ V)
18 nfv 1915 . . . 4 𝑥 𝐴 ∈ V
19 nfcv 2979 . . . 4 𝑥𝐴
20 nfcv 2979 . . . . 5 𝑥(EndoFMnd‘𝐴)
21 nfcv 2979 . . . . 5 𝑥s
22 nfab1 2981 . . . . . 6 𝑥{𝑥𝑥:𝐴1-1-onto𝐴}
2312, 22nfcxfr 2977 . . . . 5 𝑥𝐵
2420, 21, 23nfov 7170 . . . 4 𝑥((EndoFMnd‘𝐴) ↾s 𝐵)
253, 15, 16, 17, 18, 19, 24fvmptdf 6756 . . 3 (𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵))
26 ress0 16549 . . . . 5 (∅ ↾s 𝐵) = ∅
2726a1i 11 . . . 4 𝐴 ∈ V → (∅ ↾s 𝐵) = ∅)
28 fvprc 6645 . . . . 5 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅)
2928oveq1d 7155 . . . 4 𝐴 ∈ V → ((EndoFMnd‘𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
30 fvprc 6645 . . . 4 𝐴 ∈ V → (SymGrp‘𝐴) = ∅)
3127, 29, 303eqtr4rd 2868 . . 3 𝐴 ∈ V → (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵))
3225, 31pm2.61i 185 . 2 (SymGrp‘𝐴) = ((EndoFMnd‘𝐴) ↾s 𝐵)
331, 32eqtri 2845 1 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2114  {cab 2800  Vcvv 3469  ∅c0 4265   ↦ cmpt 5122  –1-1-onto→wf1o 6333  ‘cfv 6334  (class class class)co 7140   ↾s cress 16475  EndoFMndcefmnd 18024  SymGrpcsymg 18486 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-slot 16478  df-base 16480  df-ress 16482  df-symg 18487 This theorem is referenced by:  symgbas  18490  symgressbas  18501  symgplusg  18502  symgvalstruct  18516  symgtset  18518
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