Theorem fseqen 9641

Description: A set that is equinumerous to its Cartesian product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.)

Assertion

Ref	Expression
fseqen	⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))

Distinct variable group:   𝐴,𝑛

Proof of Theorem fseqen

Dummy variables 𝑓 𝑏 𝑔 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8636	. 2 ⊢ ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
2		n0 4261	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐴)
3		exdistrv 1964	. . 3 ⊢ (∃𝑓∃𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) ↔ (∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐴))
4		omex 9258	. . . . . . 7 ⊢ ω ∈ V
5		simpl 486	. . . . . . . . 9 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
6		f1ofo 6668	. . . . . . . . 9 ⊢ (𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 → 𝑓:(𝐴 × 𝐴)–onto→𝐴)
7		forn 6636	. . . . . . . . 9 ⊢ (𝑓:(𝐴 × 𝐴)–onto→𝐴 → ran 𝑓 = 𝐴)
8	5, 6, 7	3syl 18	. . . . . . . 8 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ran 𝑓 = 𝐴)
9		vex 3412	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10	9	rnex 7690	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
11	8, 10	eqeltrrdi 2847	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝐴 ∈ V)
12		xpexg 7535	. . . . . . 7 ⊢ ((ω ∈ V ∧ 𝐴 ∈ V) → (ω × 𝐴) ∈ V)
13	4, 11, 12	sylancr 590	. . . . . 6 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → (ω × 𝐴) ∈ V)
14		simpr 488	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
15		eqid 2737	. . . . . . 7 ⊢ seqω((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴 ↑m suc 𝑘) ↦ ((𝑔‘(𝑦 ↾ 𝑘))𝑓(𝑦‘𝑘)))), {⟨∅, 𝑏⟩}) = seqω((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴 ↑m suc 𝑘) ↦ ((𝑔‘(𝑦 ↾ 𝑘))𝑓(𝑦‘𝑘)))), {⟨∅, 𝑏⟩})
16		eqid 2737	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ⟨dom 𝑥, ((seqω((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴 ↑m suc 𝑘) ↦ ((𝑔‘(𝑦 ↾ 𝑘))𝑓(𝑦‘𝑘)))), {⟨∅, 𝑏⟩})‘dom 𝑥)‘𝑥)⟩) = (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ⟨dom 𝑥, ((seqω((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴 ↑m suc 𝑘) ↦ ((𝑔‘(𝑦 ↾ 𝑘))𝑓(𝑦‘𝑘)))), {⟨∅, 𝑏⟩})‘dom 𝑥)‘𝑥)⟩)
17	11, 14, 5, 15, 16	fseqenlem2 9639	. . . . . 6 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ⟨dom 𝑥, ((seqω((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴 ↑m suc 𝑘) ↦ ((𝑔‘(𝑦 ↾ 𝑘))𝑓(𝑦‘𝑘)))), {⟨∅, 𝑏⟩})‘dom 𝑥)‘𝑥)⟩):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)–1-1→(ω × 𝐴))
18		f1domg 8648	. . . . . 6 ⊢ ((ω × 𝐴) ∈ V → ((𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ⟨dom 𝑥, ((seqω((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴 ↑m suc 𝑘) ↦ ((𝑔‘(𝑦 ↾ 𝑘))𝑓(𝑦‘𝑘)))), {⟨∅, 𝑏⟩})‘dom 𝑥)‘𝑥)⟩):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)–1-1→(ω × 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ (ω × 𝐴)))
19	13, 17, 18	sylc 65	. . . . 5 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ (ω × 𝐴))
20		fseqdom 9640	. . . . . 6 ⊢ (𝐴 ∈ V → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
21	11, 20	syl 17	. . . . 5 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
22		sbth 8766	. . . . 5 ⊢ ((∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ (ω × 𝐴) ∧ (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
23	19, 21, 22	syl2anc 587	. . . 4 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
24	23	exlimivv 1940	. . 3 ⊢ (∃𝑓∃𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
25	3, 24	sylbir 238	. 2 ⊢ ((∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
26	1, 2, 25	syl2anb 601	1 ⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2110   ≠ wne 2940  Vcvv 3408  ∅c0 4237  {csn 4541  ⟨cop 4547  ∪ ciun 4904   class class class wbr 5053   ↦ cmpt 5135   × cxp 5549  dom cdm 5551  ran crn 5552   ↾ cres 5553  suc csuc 6215  –1-1→wf1 6377  –onto→wfo 6378  –1-1-onto→wf1o 6379  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  ωcom 7644  seq_ωcseqom 8183   ↑_m cmap 8508   ≈ cen 8623   ≼ cdom 8624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-seqom 8184  df-1o 8202  df-map 8510  df-en 8627  df-dom 8628

This theorem is referenced by:  infpwfien  9676