Theorem fseqen 9783

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 8743	. 2 ⊢ ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
2		n0 4280	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐴)
3		exdistrv 1959	. . 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 9401	. . . . . . 7 ⊢ ω ∈ V
5		simpl 483	. . . . . . . . 9 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
6		f1ofo 6723	. . . . . . . . 9 ⊢ (𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 → 𝑓:(𝐴 × 𝐴)–onto→𝐴)
7		forn 6691	. . . . . . . . 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 3436	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10	9	rnex 7759	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
11	8, 10	eqeltrrdi 2848	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝐴 ∈ V)
12		xpexg 7600	. . . . . . 7 ⊢ ((ω ∈ V ∧ 𝐴 ∈ V) → (ω × 𝐴) ∈ V)
13	4, 11, 12	sylancr 587	. . . . . 6 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → (ω × 𝐴) ∈ V)
14		simpr 485	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
15		eqid 2738	. . . . . . 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 2738	. . . . . . 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 9781	. . . . . 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 8760	. . . . . 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 9782	. . . . . 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 8880	. . . . 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 584	. . . 4 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
24	23	exlimivv 1935	. . 3 ⊢ (∃𝑓∃𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
25	3, 24	sylbir 234	. 2 ⊢ ((∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
26	1, 2, 25	syl2anb 598	1 ⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432  ∅c0 4256  {csn 4561  ⟨cop 4567  ∪ ciun 4924   class class class wbr 5074   ↦ cmpt 5157   × cxp 5587  dom cdm 5589  ran crn 5590   ↾ cres 5591  suc csuc 6268  –1-1→wf1 6430  –onto→wfo 6431  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ωcom 7712  seq_ωcseqom 8278   ↑_m cmap 8615   ≈ cen 8730   ≼ cdom 8731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279  df-1o 8297  df-map 8617  df-en 8734  df-dom 8735

This theorem is referenced by:  infpwfien  9818