Theorem fseqen 10034

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 8966	. 2 ⊢ ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
2		n0 4303	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐴)
3		exdistrv 1988	. . 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 9626	. . . . . . 7 ⊢ ω ∈ V
5		simpl 488	. . . . . . . . 9 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
6		f1ofo 6829	. . . . . . . . 9 ⊢ (𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 → 𝑓:(𝐴 × 𝐴)–onto→𝐴)
7		forn 6796	. . . . . . . . 9 ⊢ (𝑓:(𝐴 × 𝐴)–onto→𝐴 → ran 𝑓 = 𝐴)
8	5, 6, 7	3syl 19	. . . . . . . 8 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ran 𝑓 = 𝐴)
9		vex 3457	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10	9	rnex 7911	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
11	8, 10	eqeltrrdi 2871	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝐴 ∈ V)
12		xpexg 7753	. . . . . . 7 ⊢ ((ω ∈ V ∧ 𝐴 ∈ V) → (ω × 𝐴) ∈ V)
13	4, 11, 12	sylancr 599	. . . . . 6 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → (ω × 𝐴) ∈ V)
14		simpr 490	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
15		eqid 2762	. . . . . . 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 2762	. . . . . . 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 10032	. . . . . 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 8981	. . . . . 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 66	. . . . 5 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ (ω × 𝐴))
20		fseqdom 10033	. . . . . 6 ⊢ (𝐴 ∈ V → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
21	11, 20	syl 18	. . . . 5 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
22		sbth 9099	. . . . 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 596	. . . 4 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
24	23	exlimivv 1965	. . 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 610	1 ⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  Vcvv 3453  ∅c0 4282  {csn 4587  ⟨cop 4593  ∪ ciun 4954   class class class wbr 5107   ↦ cmpt 5190   × cxp 5657  dom cdm 5659  ran crn 5660   ↾ cres 5661  suc csuc 6363  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  ωcom 7866  seq_ωcseqom 8440   ↑_m cmap 8830   ≈ cen 8953   ≼ cdom 8954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-seqom 8441  df-1o 8459  df-map 8832  df-en 8957  df-dom 8958

This theorem is used by:  infpwfien  10069