MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  symgval Structured version   Visualization version   GIF version

Theorem symgval 19277
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 19276 . . . . 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 6890 . . . . . 6 (π‘₯ = 𝐴 β†’ (EndoFMndβ€˜π‘₯) = (EndoFMndβ€˜π΄))
5 eqidd 2731 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ β„Ž = β„Ž)
6 id 22 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ π‘₯ = 𝐴)
75, 6, 6f1oeq123d 6826 . . . . . . . . 9 (π‘₯ = 𝐴 β†’ (β„Ž:π‘₯–1-1-ontoβ†’π‘₯ ↔ β„Ž:𝐴–1-1-onto→𝐴))
87abbidv 2799 . . . . . . . 8 (π‘₯ = 𝐴 β†’ {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯} = {β„Ž ∣ β„Ž:𝐴–1-1-onto→𝐴})
9 f1oeq1 6820 . . . . . . . . 9 (β„Ž = π‘₯ β†’ (β„Ž:𝐴–1-1-onto→𝐴 ↔ π‘₯:𝐴–1-1-onto→𝐴))
109cbvabv 2803 . . . . . . . 8 {β„Ž ∣ β„Ž:𝐴–1-1-onto→𝐴} = {π‘₯ ∣ π‘₯:𝐴–1-1-onto→𝐴}
118, 10eqtrdi 2786 . . . . . . 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 2788 . . . . . 6 (π‘₯ = 𝐴 β†’ {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯} = 𝐡)
144, 13oveq12d 7429 . . . . 5 (π‘₯ = 𝐴 β†’ ((EndoFMndβ€˜π‘₯) β†Ύs {β„Ž ∣ β„Ž:π‘₯–1-1-ontoβ†’π‘₯}) = ((EndoFMndβ€˜π΄) β†Ύs 𝐡))
1514adantl 480 . . . 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 7446 . . . 4 (𝐴 ∈ V β†’ ((EndoFMndβ€˜π΄) β†Ύs 𝐡) ∈ V)
18 nfv 1915 . . . 4 β„²π‘₯ 𝐴 ∈ V
19 nfcv 2901 . . . 4 β„²π‘₯𝐴
20 nfcv 2901 . . . . 5 β„²π‘₯(EndoFMndβ€˜π΄)
21 nfcv 2901 . . . . 5 β„²π‘₯ β†Ύs
22 nfab1 2903 . . . . . 6 β„²π‘₯{π‘₯ ∣ π‘₯:𝐴–1-1-onto→𝐴}
2312, 22nfcxfr 2899 . . . . 5 β„²π‘₯𝐡
2420, 21, 23nfov 7441 . . . 4 β„²π‘₯((EndoFMndβ€˜π΄) β†Ύs 𝐡)
253, 15, 16, 17, 18, 19, 24fvmptdf 7003 . . 3 (𝐴 ∈ V β†’ (SymGrpβ€˜π΄) = ((EndoFMndβ€˜π΄) β†Ύs 𝐡))
26 ress0 17192 . . . . 5 (βˆ… β†Ύs 𝐡) = βˆ…
2726a1i 11 . . . 4 (Β¬ 𝐴 ∈ V β†’ (βˆ… β†Ύs 𝐡) = βˆ…)
28 fvprc 6882 . . . . 5 (Β¬ 𝐴 ∈ V β†’ (EndoFMndβ€˜π΄) = βˆ…)
2928oveq1d 7426 . . . 4 (Β¬ 𝐴 ∈ V β†’ ((EndoFMndβ€˜π΄) β†Ύs 𝐡) = (βˆ… β†Ύs 𝐡))
30 fvprc 6882 . . . 4 (Β¬ 𝐴 ∈ V β†’ (SymGrpβ€˜π΄) = βˆ…)
3127, 29, 303eqtr4rd 2781 . . 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 2758 1 𝐺 = ((EndoFMndβ€˜π΄) β†Ύs 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1539   ∈ wcel 2104  {cab 2707  Vcvv 3472  βˆ…c0 4321   ↦ cmpt 5230  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411   β†Ύs cress 17177  EndoFMndcefmnd 18785  SymGrpcsymg 19275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-nn 12217  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-symg 19276
This theorem is referenced by:  symgbas  19279  symgressbas  19290  symgplusg  19291  symgvalstruct  19305  symgvalstructOLD  19306  symgtset  19308
  Copyright terms: Public domain W3C validator