Theorem fseqen 10041

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 8969	. 2 ⊢ ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
2		n0 4328	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐴)
3		exdistrv 1955	. . 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 9657	. . . . . . 7 ⊢ ω ∈ V
5		simpl 482	. . . . . . . . 9 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
6		f1ofo 6825	. . . . . . . . 9 ⊢ (𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 → 𝑓:(𝐴 × 𝐴)–onto→𝐴)
7		forn 6793	. . . . . . . . 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 3463	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10	9	rnex 7906	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
11	8, 10	eqeltrrdi 2843	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝐴 ∈ V)
12		xpexg 7744	. . . . . . 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 484	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
15		eqid 2735	. . . . . . 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 2735	. . . . . . 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 10039	. . . . . 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 8986	. . . . . 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 10040	. . . . . 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 9107	. . . . 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 1932	. . 3 ⊢ (∃𝑓∃𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
25	3, 24	sylbir 235	. 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 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2932  Vcvv 3459  ∅c0 4308  {csn 4601  ⟨cop 4607  ∪ ciun 4967   class class class wbr 5119   ↦ cmpt 5201   × cxp 5652  dom cdm 5654  ran crn 5655   ↾ cres 5656  suc csuc 6354  –1-1→wf1 6528  –onto→wfo 6529  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  ωcom 7861  seq_ωcseqom 8461   ↑_m cmap 8840   ≈ cen 8956   ≼ cdom 8957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-seqom 8462  df-1o 8480  df-map 8842  df-en 8960  df-dom 8961

This theorem is referenced by:  infpwfien  10076