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Theorem symgval 17622
 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 3185 . . 3 (𝐴𝑉𝐴 ∈ V)
3 ovex 6577 . . . . . . 7 (𝑎𝑚 𝑎) ∈ V
4 f1of 6050 . . . . . . . . 9 (𝑥:𝑎1-1-onto𝑎𝑥:𝑎𝑎)
5 vex 3176 . . . . . . . . . 10 𝑎 ∈ V
65, 5elmap 7772 . . . . . . . . 9 (𝑥 ∈ (𝑎𝑚 𝑎) ↔ 𝑥:𝑎𝑎)
74, 6sylibr 223 . . . . . . . 8 (𝑥:𝑎1-1-onto𝑎𝑥 ∈ (𝑎𝑚 𝑎))
87abssi 3640 . . . . . . 7 {𝑥𝑥:𝑎1-1-onto𝑎} ⊆ (𝑎𝑚 𝑎)
93, 8ssexi 4731 . . . . . 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 6043 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑎 = 𝐴) → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1312anidms 675 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1413abbidv 2728 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = {𝑥𝑥:𝐴1-1-onto𝐴})
15 symgval.2 . . . . . . . . 9 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
1614, 15syl6eqr 2662 . . . . . . . 8 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = 𝐵)
1711, 16sylan9eqr 2666 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑏 = 𝐵)
1817opeq2d 4347 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
19 eqidd 2611 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑔) = (𝑓𝑔))
2017, 17, 19mpt2eq123dv 6615 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔)))
21 symgval.3 . . . . . . . 8 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
2220, 21syl6eqr 2662 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = + )
2322opeq2d 4347 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
24 simpl 472 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑎 = 𝐴)
2524pweqd 4113 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝒫 𝑎 = 𝒫 𝐴)
2625sneqd 4137 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → {𝒫 𝑎} = {𝒫 𝐴})
2724, 26xpeq12d 5064 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
2827fveq2d 6107 . . . . . . . 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 2662 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
3130opeq2d 4347 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
3218, 23, 31tpeq123d 4227 . . . . 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 3526 . . . 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 17621 . . . 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 6855 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
3633, 34, 35fvmpt 6191 . . 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 2656 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  ⦋csb 3499  𝒫 cpw 4108  {csn 4125  {ctp 4129  ⟨cop 4131   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  ndxcnx 15692  Basecbs 15695  +gcplusg 15768  TopSetcts 15774  ∏tcpt 15922  SymGrpcsymg 17620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-symg 17621 This theorem is referenced by:  symgbas  17623  symgplusg  17632  symgtset  17642
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