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Theorem symgval 18006
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 3364 . . 3 (𝐴𝑉𝐴 ∈ V)
3 ovex 6823 . . . . . . 7 (𝑎𝑚 𝑎) ∈ V
4 f1of 6278 . . . . . . . . 9 (𝑥:𝑎1-1-onto𝑎𝑥:𝑎𝑎)
5 vex 3354 . . . . . . . . . 10 𝑎 ∈ V
65, 5elmap 8038 . . . . . . . . 9 (𝑥 ∈ (𝑎𝑚 𝑎) ↔ 𝑥:𝑎𝑎)
74, 6sylibr 224 . . . . . . . 8 (𝑥:𝑎1-1-onto𝑎𝑥 ∈ (𝑎𝑚 𝑎))
87abssi 3826 . . . . . . 7 {𝑥𝑥:𝑎1-1-onto𝑎} ⊆ (𝑎𝑚 𝑎)
93, 8ssexi 4937 . . . . . 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 6271 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑎 = 𝐴) → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1312anidms 556 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1413abbidv 2890 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = {𝑥𝑥:𝐴1-1-onto𝐴})
15 symgval.2 . . . . . . . . 9 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
1614, 15syl6eqr 2823 . . . . . . . 8 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = 𝐵)
1711, 16sylan9eqr 2827 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑏 = 𝐵)
1817opeq2d 4546 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
19 eqidd 2772 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑔) = (𝑓𝑔))
2017, 17, 19mpt2eq123dv 6864 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔)))
21 symgval.3 . . . . . . . 8 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
2220, 21syl6eqr 2823 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = + )
2322opeq2d 4546 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
24 simpl 468 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑎 = 𝐴)
2524pweqd 4302 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝒫 𝑎 = 𝒫 𝐴)
2625sneqd 4328 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → {𝒫 𝑎} = {𝒫 𝐴})
2724, 26xpeq12d 5280 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
2827fveq2d 6336 . . . . . . . 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 2823 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
3130opeq2d 4546 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
3218, 23, 31tpeq123d 4419 . . . . 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 3709 . . . 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 18005 . . . 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 7104 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
3633, 34, 35fvmpt 6424 . . 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 2817 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  Vcvv 3351  csb 3682  𝒫 cpw 4297  {csn 4316  {ctp 4320  cop 4322   × cxp 5247  ccom 5253  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  cmpt2 6795  𝑚 cmap 8009  ndxcnx 16061  Basecbs 16064  +gcplusg 16149  TopSetcts 16155  tcpt 16307  SymGrpcsymg 18004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-symg 18005
This theorem is referenced by:  symgbas  18007  symgplusg  18016  symgtset  18026
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