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 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
symgval.3	⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
symgval.4	⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))

Assertion

Ref	Expression
symgval	⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(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.

Step	Hyp	Ref	Expression
1		symgval.1	. 2 ⊢ 𝐺 = (SymGrp‘𝐴)
2		elex 3417	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		ovex 6916	. . . . . . 7 ⊢ (𝑎 ↑𝑚 𝑎) ∈ V
4		f1of 6363	. . . . . . . . 9 ⊢ (𝑥:𝑎–1-1-onto→𝑎 → 𝑥:𝑎⟶𝑎)
5		vex 3405	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
6	5, 5	elmap 8131	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ↑𝑚 𝑎) ↔ 𝑥:𝑎⟶𝑎)
7	4, 6	sylibr 225	. . . . . . . 8 ⊢ (𝑥:𝑎–1-1-onto→𝑎 → 𝑥 ∈ (𝑎 ↑𝑚 𝑎))
8	7	abssi 3885	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} ⊆ (𝑎 ↑𝑚 𝑎)
9	3, 8	ssexi 5011	. . . . . 6 ⊢ {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} ∈ V
10	9	a1i 11	. . . . 5 ⊢ (𝑎 = 𝐴 → {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} ∈ V)
11		id 22	. . . . . . . 8 ⊢ (𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} → 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎})
12		f1oeq23 6356	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝐴 ∧ 𝑎 = 𝐴) → (𝑥:𝑎–1-1-onto→𝑎 ↔ 𝑥:𝐴–1-1-onto→𝐴))
13	12	anidms 558	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑥:𝑎–1-1-onto→𝑎 ↔ 𝑥:𝐴–1-1-onto→𝐴))
14	13	abbidv 2936	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴})
15		symgval.2	. . . . . . . . 9 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
16	14, 15	syl6eqr 2869	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} = 𝐵)
17	11, 16	sylan9eqr 2873	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → 𝑏 = 𝐵)
18	17	opeq2d 4613	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
19		eqidd 2818	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
20	17, 17, 19	mpt2eq123dv 6957	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)))
21		symgval.3	. . . . . . . 8 ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
22	20, 21	syl6eqr 2869	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = + )
23	22	opeq2d 4613	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
24		simpl 470	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → 𝑎 = 𝐴)
25	24	pweqd 4367	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → 𝒫 𝑎 = 𝒫 𝐴)
26	25	sneqd 4393	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → {𝒫 𝑎} = {𝒫 𝐴})
27	24, 26	xpeq12d 5354	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
28	27	fveq2d 6422	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = (∏t‘(𝐴 × {𝒫 𝐴})))
29		symgval.4	. . . . . . . 8 ⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
30	28, 29	syl6eqr 2869	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
31	30	opeq2d 4613	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
32	18, 23, 31	tpeq123d 4485	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
33	10, 32	csbied 3766	. . . 4 ⊢ (𝑎 = 𝐴 → ⦋{𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
34		df-symg 18019	. . . 4 ⊢ SymGrp = (𝑎 ∈ V ↦ ⦋{𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩})
35		tpex 7197	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
36	33, 34, 35	fvmpt 6513	. . 3 ⊢ (𝐴 ∈ V → (SymGrp‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
37	2, 36	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (SymGrp‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
38	1, 37	syl5eq 2863	1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

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-onto→wf1o 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