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Theorem symgval 17999
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 3352 . . 3 (𝐴𝑉𝐴 ∈ V)
3 ovex 6841 . . . . . . 7 (𝑎𝑚 𝑎) ∈ V
4 f1of 6298 . . . . . . . . 9 (𝑥:𝑎1-1-onto𝑎𝑥:𝑎𝑎)
5 vex 3343 . . . . . . . . . 10 𝑎 ∈ V
65, 5elmap 8052 . . . . . . . . 9 (𝑥 ∈ (𝑎𝑚 𝑎) ↔ 𝑥:𝑎𝑎)
74, 6sylibr 224 . . . . . . . 8 (𝑥:𝑎1-1-onto𝑎𝑥 ∈ (𝑎𝑚 𝑎))
87abssi 3818 . . . . . . 7 {𝑥𝑥:𝑎1-1-onto𝑎} ⊆ (𝑎𝑚 𝑎)
93, 8ssexi 4955 . . . . . 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 6291 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑎 = 𝐴) → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1312anidms 680 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1413abbidv 2879 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = {𝑥𝑥:𝐴1-1-onto𝐴})
15 symgval.2 . . . . . . . . 9 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
1614, 15syl6eqr 2812 . . . . . . . 8 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = 𝐵)
1711, 16sylan9eqr 2816 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑏 = 𝐵)
1817opeq2d 4560 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
19 eqidd 2761 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑔) = (𝑓𝑔))
2017, 17, 19mpt2eq123dv 6882 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔)))
21 symgval.3 . . . . . . . 8 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
2220, 21syl6eqr 2812 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = + )
2322opeq2d 4560 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
24 simpl 474 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑎 = 𝐴)
2524pweqd 4307 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝒫 𝑎 = 𝒫 𝐴)
2625sneqd 4333 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → {𝒫 𝑎} = {𝒫 𝐴})
2724, 26xpeq12d 5297 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
2827fveq2d 6356 . . . . . . . 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 2812 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
3130opeq2d 4560 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
3218, 23, 31tpeq123d 4427 . . . . 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 3701 . . . 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 17998 . . . 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 7122 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
3633, 34, 35fvmpt 6444 . . 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 2806 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {cab 2746  Vcvv 3340  csb 3674  𝒫 cpw 4302  {csn 4321  {ctp 4325  cop 4327   × cxp 5264  ccom 5270  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  ndxcnx 16056  Basecbs 16059  +gcplusg 16143  TopSetcts 16149  tcpt 16301  SymGrpcsymg 17997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-symg 17998
This theorem is referenced by:  symgbas  18000  symgplusg  18009  symgtset  18019
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