Theorem symgbas 17800

Description: The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.)

Hypotheses

Ref	Expression
symgbas.1	⊢ 𝐺 = (SymGrp‘𝐴)
symgbas.2	⊢ 𝐵 = (Base‘𝐺)

Assertion

Ref	Expression
symgbas	⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐺(𝑥)

Proof of Theorem symgbas

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		symgbas.2	. 2 ⊢ 𝐵 = (Base‘𝐺)
2		f1of 6137	. . . . . . . 8 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → 𝑥:𝐴⟶𝐴)
3		elmapg 7870	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐴 ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶𝐴))
4	3	anidms 677	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐴 ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶𝐴))
5	2, 4	syl5ibr 236	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝑥:𝐴–1-1-onto→𝐴 → 𝑥 ∈ (𝐴 ↑𝑚 𝐴)))
6	5	abssdv 3676	. . . . . 6 ⊢ (𝐴 ∈ V → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ⊆ (𝐴 ↑𝑚 𝐴))
7		ovex 6678	. . . . . 6 ⊢ (𝐴 ↑𝑚 𝐴) ∈ V
8		ssexg 4804	. . . . . 6 ⊢ (({𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ⊆ (𝐴 ↑𝑚 𝐴) ∧ (𝐴 ↑𝑚 𝐴) ∈ V) → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ∈ V)
9	6, 7, 8	sylancl 694	. . . . 5 ⊢ (𝐴 ∈ V → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ∈ V)
10		eqid 2622	. . . . . 6 ⊢ {⟨(Base‘ndx), {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}⟩, ⟨(+g‘ndx), (𝑓 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}, 𝑔 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩} = {⟨(Base‘ndx), {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}⟩, ⟨(+g‘ndx), (𝑓 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}, 𝑔 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩}
11	10	topgrpbas 16043	. . . . 5 ⊢ ({𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ∈ V → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = (Base‘{⟨(Base‘ndx), {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}⟩, ⟨(+g‘ndx), (𝑓 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}, 𝑔 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩}))
12	9, 11	syl 17	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = (Base‘{⟨(Base‘ndx), {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}⟩, ⟨(+g‘ndx), (𝑓 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}, 𝑔 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩}))
13		symgbas.1	. . . . . 6 ⊢ 𝐺 = (SymGrp‘𝐴)
14		eqid 2622	. . . . . 6 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
15		eqid 2622	. . . . . 6 ⊢ (𝑓 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}, 𝑔 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}, 𝑔 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ↦ (𝑓 ∘ 𝑔))
16		eqid 2622	. . . . . 6 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
17	13, 14, 15, 16	symgval 17799	. . . . 5 ⊢ (𝐴 ∈ V → 𝐺 = {⟨(Base‘ndx), {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}⟩, ⟨(+g‘ndx), (𝑓 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}, 𝑔 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩})
18	17	fveq2d 6195	. . . 4 ⊢ (𝐴 ∈ V → (Base‘𝐺) = (Base‘{⟨(Base‘ndx), {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}⟩, ⟨(+g‘ndx), (𝑓 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}, 𝑔 ∈ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩}))
19	12, 18	eqtr4d 2659	. . 3 ⊢ (𝐴 ∈ V → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = (Base‘𝐺))
20		base0 15912	. . . 4 ⊢ ∅ = (Base‘∅)
21		f1odm 6141	. . . . . . . . 9 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → dom 𝑥 = 𝐴)
22		vex 3203	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
23	22	dmex 7099	. . . . . . . . 9 ⊢ dom 𝑥 ∈ V
24	21, 23	syl6eqelr 2710	. . . . . . . 8 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → 𝐴 ∈ V)
25	24	con3i 150	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ¬ 𝑥:𝐴–1-1-onto→𝐴)
26	25	pm2.21d 118	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑥:𝐴–1-1-onto→𝐴 → 𝑥 ∈ ∅))
27	26	abssdv 3676	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ⊆ ∅)
28		ss0 3974	. . . . 5 ⊢ ({𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ⊆ ∅ → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = ∅)
29	27, 28	syl 17	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = ∅)
30		fvprc 6185	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅)
31	13, 30	syl5eq 2668	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐺 = ∅)
32	31	fveq2d 6195	. . . 4 ⊢ (¬ 𝐴 ∈ V → (Base‘𝐺) = (Base‘∅))
33	20, 29, 32	3eqtr4a 2682	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = (Base‘𝐺))
34	19, 33	pm2.61i 176	. 2 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = (Base‘𝐺)
35	1, 34	eqtr4i 2647	1 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}

Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  {ctp 4181  ⟨cop 4183   × cxp 5112  dom cdm 5114   ∘ 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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-pss 3590  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-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-tset 15960  df-symg 17798

This theorem is referenced by:  elsymgbas2  17801  symghash  17805  symgbasfi  17806  symgplusg  17809  symgbas0  17814  symg1bas  17816  symgtset  17819