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Theorem symgval 19278
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 19277 . . . . 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 2732 . . . . . . . . . 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 2800 . . . . . . . 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 2804 . . . . . . . 8 {β„Ž ∣ β„Ž:𝐴–1-1-onto→𝐴} = {π‘₯ ∣ π‘₯:𝐴–1-1-onto→𝐴}
118, 10eqtrdi 2787 . . . . . . 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 2789 . . . . . 6 (π‘₯ = 𝐴 β†’ {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯} = 𝐡)
144, 13oveq12d 7430 . . . . 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 7447 . . . 4 (𝐴 ∈ V β†’ ((EndoFMndβ€˜π΄) β†Ύs 𝐡) ∈ V)
18 nfv 1916 . . . 4 β„²π‘₯ 𝐴 ∈ V
19 nfcv 2902 . . . 4 β„²π‘₯𝐴
20 nfcv 2902 . . . . 5 β„²π‘₯(EndoFMndβ€˜π΄)
21 nfcv 2902 . . . . 5 β„²π‘₯ β†Ύs
22 nfab1 2904 . . . . . 6 β„²π‘₯{π‘₯ ∣ π‘₯:𝐴–1-1-onto→𝐴}
2312, 22nfcxfr 2900 . . . . 5 β„²π‘₯𝐡
2420, 21, 23nfov 7442 . . . 4 β„²π‘₯((EndoFMndβ€˜π΄) β†Ύs 𝐡)
253, 15, 16, 17, 18, 19, 24fvmptdf 7004 . . 3 (𝐴 ∈ V β†’ (SymGrpβ€˜π΄) = ((EndoFMndβ€˜π΄) β†Ύs 𝐡))
26 ress0 17193 . . . . 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 7427 . . . 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 2782 . . 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 2759 1 𝐺 = ((EndoFMndβ€˜π΄) β†Ύs 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1540   ∈ wcel 2105  {cab 2708  Vcvv 3473  βˆ…c0 4322   ↦ cmpt 5231  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7412   β†Ύs cress 17178  EndoFMndcefmnd 18786  SymGrpcsymg 19276
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-cnex 11169  ax-1cn 11171  ax-addcl 11173
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-nn 12218  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-symg 19277
This theorem is referenced by:  symgbas  19280  symgressbas  19291  symgplusg  19292  symgvalstruct  19306  symgvalstructOLD  19307  symgtset  19309
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