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Theorem symgval 19314
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 = (SymGrpβ€˜π΄)
symgval.2 𝐡 = {π‘₯ ∣ π‘₯:𝐴–1-1-onto→𝐴}
Assertion
Ref Expression
symgval 𝐺 = ((EndoFMndβ€˜π΄) β†Ύs 𝐡)
Distinct variable group:   π‘₯,𝐴
Allowed substitution hints:   𝐡(π‘₯)   𝐺(π‘₯)

Proof of Theorem symgval
Dummy variable β„Ž is distinct from all other variables.
StepHypRef Expression
1 symgval.1 . 2 𝐺 = (SymGrpβ€˜π΄)
2 df-symg 19313 . . . . 5 SymGrp = (π‘₯ ∈ V ↦ ((EndoFMndβ€˜π‘₯) β†Ύs {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯}))
32a1i 11 . . . 4 (𝐴 ∈ V β†’ SymGrp = (π‘₯ ∈ V ↦ ((EndoFMndβ€˜π‘₯) β†Ύs {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯})))
4 fveq2 6891 . . . . . 6 (π‘₯ = 𝐴 β†’ (EndoFMndβ€˜π‘₯) = (EndoFMndβ€˜π΄))
5 eqidd 2728 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ β„Ž = β„Ž)
6 id 22 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ π‘₯ = 𝐴)
75, 6, 6f1oeq123d 6827 . . . . . . . . 9 (π‘₯ = 𝐴 β†’ (β„Ž:π‘₯–1-1-ontoβ†’π‘₯ ↔ β„Ž:𝐴–1-1-onto→𝐴))
87abbidv 2796 . . . . . . . 8 (π‘₯ = 𝐴 β†’ {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯} = {β„Ž ∣ β„Ž:𝐴–1-1-onto→𝐴})
9 f1oeq1 6821 . . . . . . . . 9 (β„Ž = π‘₯ β†’ (β„Ž:𝐴–1-1-onto→𝐴 ↔ π‘₯:𝐴–1-1-onto→𝐴))
109cbvabv 2800 . . . . . . . 8 {β„Ž ∣ β„Ž:𝐴–1-1-onto→𝐴} = {π‘₯ ∣ π‘₯:𝐴–1-1-onto→𝐴}
118, 10eqtrdi 2783 . . . . . . 7 (π‘₯ = 𝐴 β†’ {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯} = {π‘₯ ∣ π‘₯:𝐴–1-1-onto→𝐴})
12 symgval.2 . . . . . . 7 𝐡 = {π‘₯ ∣ π‘₯:𝐴–1-1-onto→𝐴}
1311, 12eqtr4di 2785 . . . . . 6 (π‘₯ = 𝐴 β†’ {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯} = 𝐡)
144, 13oveq12d 7432 . . . . 5 (π‘₯ = 𝐴 β†’ ((EndoFMndβ€˜π‘₯) β†Ύs {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯}) = ((EndoFMndβ€˜π΄) β†Ύs 𝐡))
1514adantl 481 . . . 4 ((𝐴 ∈ V ∧ π‘₯ = 𝐴) β†’ ((EndoFMndβ€˜π‘₯) β†Ύs {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯}) = ((EndoFMndβ€˜π΄) β†Ύs 𝐡))
16 id 22 . . . 4 (𝐴 ∈ V β†’ 𝐴 ∈ V)
17 ovexd 7449 . . . 4 (𝐴 ∈ V β†’ ((EndoFMndβ€˜π΄) β†Ύs 𝐡) ∈ V)
18 nfv 1910 . . . 4 β„²π‘₯ 𝐴 ∈ V
19 nfcv 2898 . . . 4 β„²π‘₯𝐴
20 nfcv 2898 . . . . 5 β„²π‘₯(EndoFMndβ€˜π΄)
21 nfcv 2898 . . . . 5 β„²π‘₯ β†Ύs
22 nfab1 2900 . . . . . 6 β„²π‘₯{π‘₯ ∣ π‘₯:𝐴–1-1-onto→𝐴}
2312, 22nfcxfr 2896 . . . . 5 β„²π‘₯𝐡
2420, 21, 23nfov 7444 . . . 4 β„²π‘₯((EndoFMndβ€˜π΄) β†Ύs 𝐡)
253, 15, 16, 17, 18, 19, 24fvmptdf 7005 . . 3 (𝐴 ∈ V β†’ (SymGrpβ€˜π΄) = ((EndoFMndβ€˜π΄) β†Ύs 𝐡))
26 ress0 17215 . . . . 5 (βˆ… β†Ύs 𝐡) = βˆ…
2726a1i 11 . . . 4 (Β¬ 𝐴 ∈ V β†’ (βˆ… β†Ύs 𝐡) = βˆ…)
28 fvprc 6883 . . . . 5 (Β¬ 𝐴 ∈ V β†’ (EndoFMndβ€˜π΄) = βˆ…)
2928oveq1d 7429 . . . 4 (Β¬ 𝐴 ∈ V β†’ ((EndoFMndβ€˜π΄) β†Ύs 𝐡) = (βˆ… β†Ύs 𝐡))
30 fvprc 6883 . . . 4 (Β¬ 𝐴 ∈ V β†’ (SymGrpβ€˜π΄) = βˆ…)
3127, 29, 303eqtr4rd 2778 . . 3 (Β¬ 𝐴 ∈ V β†’ (SymGrpβ€˜π΄) = ((EndoFMndβ€˜π΄) β†Ύs 𝐡))
3225, 31pm2.61i 182 . 2 (SymGrpβ€˜π΄) = ((EndoFMndβ€˜π΄) β†Ύs 𝐡)
331, 32eqtri 2755 1 𝐺 = ((EndoFMndβ€˜π΄) β†Ύs 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1534   ∈ wcel 2099  {cab 2704  Vcvv 3469  βˆ…c0 4318   ↦ cmpt 5225  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414   β†Ύs cress 17200  EndoFMndcefmnd 18811  SymGrpcsymg 19312
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 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-1cn 11188  ax-addcl 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-nn 12235  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-symg 19313
This theorem is referenced by:  symgbas  19316  symgressbas  19327  symgplusg  19328  symgvalstruct  19342  symgvalstructOLD  19343  symgtset  19345
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