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Theorem symgval 18020
Description: The value of the symmetric group function at 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.)
Hypotheses
Ref Expression
symgval.1 𝐺 = (SymGrp‘𝐴)
symgval.2 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
symgval.3 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
symgval.4 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
Assertion
Ref Expression
symgval (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Distinct variable group:   𝑓,𝑔,𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑓,𝑔)   + (𝑥,𝑓,𝑔)   𝐺(𝑥,𝑓,𝑔)   𝐽(𝑥,𝑓,𝑔)   𝑉(𝑥,𝑓,𝑔)

Proof of Theorem symgval
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symgval.1 . 2 𝐺 = (SymGrp‘𝐴)
2 elex 3417 . . 3 (𝐴𝑉𝐴 ∈ V)
3 ovex 6916 . . . . . . 7 (𝑎𝑚 𝑎) ∈ V
4 f1of 6363 . . . . . . . . 9 (𝑥:𝑎1-1-onto𝑎𝑥:𝑎𝑎)
5 vex 3405 . . . . . . . . . 10 𝑎 ∈ V
65, 5elmap 8131 . . . . . . . . 9 (𝑥 ∈ (𝑎𝑚 𝑎) ↔ 𝑥:𝑎𝑎)
74, 6sylibr 225 . . . . . . . 8 (𝑥:𝑎1-1-onto𝑎𝑥 ∈ (𝑎𝑚 𝑎))
87abssi 3885 . . . . . . 7 {𝑥𝑥:𝑎1-1-onto𝑎} ⊆ (𝑎𝑚 𝑎)
93, 8ssexi 5011 . . . . . 6 {𝑥𝑥:𝑎1-1-onto𝑎} ∈ V
109a1i 11 . . . . 5 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} ∈ V)
11 id 22 . . . . . . . 8 (𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎} → 𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎})
12 f1oeq23 6356 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑎 = 𝐴) → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1312anidms 558 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1413abbidv 2936 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = {𝑥𝑥:𝐴1-1-onto𝐴})
15 symgval.2 . . . . . . . . 9 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
1614, 15syl6eqr 2869 . . . . . . . 8 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = 𝐵)
1711, 16sylan9eqr 2873 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑏 = 𝐵)
1817opeq2d 4613 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
19 eqidd 2818 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑔) = (𝑓𝑔))
2017, 17, 19mpt2eq123dv 6957 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔)))
21 symgval.3 . . . . . . . 8 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
2220, 21syl6eqr 2869 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = + )
2322opeq2d 4613 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
24 simpl 470 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑎 = 𝐴)
2524pweqd 4367 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝒫 𝑎 = 𝒫 𝐴)
2625sneqd 4393 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → {𝒫 𝑎} = {𝒫 𝐴})
2724, 26xpeq12d 5354 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
2827fveq2d 6422 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = (∏t‘(𝐴 × {𝒫 𝐴})))
29 symgval.4 . . . . . . . 8 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
3028, 29syl6eqr 2869 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
3130opeq2d 4613 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
3218, 23, 31tpeq123d 4485 . . . . 5 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3310, 32csbied 3766 . . . 4 (𝑎 = 𝐴{𝑥𝑥:𝑎1-1-onto𝑎} / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
34 df-symg 18019 . . . 4 SymGrp = (𝑎 ∈ V ↦ {𝑥𝑥:𝑎1-1-onto𝑎} / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩})
35 tpex 7197 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
3633, 34, 35fvmpt 6513 . . 3 (𝐴 ∈ V → (SymGrp‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
372, 36syl 17 . 2 (𝐴𝑉 → (SymGrp‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
381, 37syl5eq 2863 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  {cab 2803  Vcvv 3402  csb 3739  𝒫 cpw 4362  {csn 4381  {ctp 4385  cop 4387   × cxp 5322  ccom 5328  wf 6107  1-1-ontowf1o 6110  cfv 6111  (class class class)co 6884  cmpt2 6886  𝑚 cmap 8102  ndxcnx 16085  Basecbs 16088  +gcplusg 16173  TopSetcts 16179  tcpt 16324  SymGrpcsymg 18018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-map 8104  df-symg 18019
This theorem is referenced by:  symgbas  18021  symgplusg  18030  symgtset  18040
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