Theorem symgval 17799

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 3212	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		ovex 6678	. . . . . . 7 ⊢ (𝑎 ↑𝑚 𝑎) ∈ V
4		f1of 6137	. . . . . . . . 9 ⊢ (𝑥:𝑎–1-1-onto→𝑎 → 𝑥:𝑎⟶𝑎)
5		vex 3203	. . . . . . . . . 10 ⊢ 𝑎 ∈ V
6	5, 5	elmap 7886	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑎 ↑𝑚 𝑎) ↔ 𝑥:𝑎⟶𝑎)
7	4, 6	sylibr 224	. . . . . . . 8 ⊢ (𝑥:𝑎–1-1-onto→𝑎 → 𝑥 ∈ (𝑎 ↑𝑚 𝑎))
8	7	abssi 3677	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} ⊆ (𝑎 ↑𝑚 𝑎)
9	3, 8	ssexi 4803	. . . . . 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 6130	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝐴 ∧ 𝑎 = 𝐴) → (𝑥:𝑎–1-1-onto→𝑎 ↔ 𝑥:𝐴–1-1-onto→𝐴))
13	12	anidms 677	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑥:𝑎–1-1-onto→𝑎 ↔ 𝑥:𝐴–1-1-onto→𝐴))
14	13	abbidv 2741	. . . . . . . . 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 2674	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎} = 𝐵)
17	11, 16	sylan9eqr 2678	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → 𝑏 = 𝐵)
18	17	opeq2d 4409	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
19		eqidd 2623	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
20	17, 17, 19	mpt2eq123dv 6717	. . . . . . . 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 2674	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = + )
23	22	opeq2d 4409	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
24		simpl 473	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → 𝑎 = 𝐴)
25	24	pweqd 4163	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → 𝒫 𝑎 = 𝒫 𝐴)
26	25	sneqd 4189	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → {𝒫 𝑎} = {𝒫 𝐴})
27	24, 26	xpeq12d 5140	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
28	27	fveq2d 6195	. . . . . . . 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 2674	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = {𝑥 ∣ 𝑥:𝑎–1-1-onto→𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
31	30	opeq2d 4409	. . . . . 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 4283	. . . . 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 3560	. . . 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 17798	. . . 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 6957	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
36	33, 34, 35	fvmpt 6282	. . 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 2668	1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200  ⦋csb 3533  𝒫 cpw 4158  {csn 4177  {ctp 4181  ⟨cop 4183   × cxp 5112   ∘ ccom 5118  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑_𝑚 cmap 7857  ndxcnx 15854  Basecbs 15857  +_gcplusg 15941  TopSetcts 15947  ∏_tcpt 16099  SymGrpcsymg 17797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-symg 17798

This theorem is referenced by:  symgbas  17800  symgplusg  17809  symgtset  17819