Theorem fseqen 9455

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 8520	. 2 ⊢ ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
2		n0 4312	. 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐴)
3		exdistrv 1956	. . 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 9108	. . . . . . 7 ⊢ ω ∈ V
5		simpl 485	. . . . . . . . 9 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴)
6		f1ofo 6624	. . . . . . . . 9 ⊢ (𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 → 𝑓:(𝐴 × 𝐴)–onto→𝐴)
7		forn 6595	. . . . . . . . 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 3499	. . . . . . . . 9 ⊢ 𝑓 ∈ V
10	9	rnex 7619	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
11	8, 10	eqeltrrdi 2924	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝐴 ∈ V)
12		xpexg 7475	. . . . . . 7 ⊢ ((ω ∈ V ∧ 𝐴 ∈ V) → (ω × 𝐴) ∈ V)
13	4, 11, 12	sylancr 589	. . . . . 6 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → (ω × 𝐴) ∈ V)
14		simpr 487	. . . . . . 7 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
15		eqid 2823	. . . . . . 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 2823	. . . . . . 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 9453	. . . . . 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 8531	. . . . . 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 9454	. . . . . 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 8639	. . . . 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 586	. . . 4 ⊢ ((𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
24	23	exlimivv 1933	. . 3 ⊢ (∃𝑓∃𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
25	3, 24	sylbir 237	. 2 ⊢ ((∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto→𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
26	1, 2, 25	syl2anb 599	1 ⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  Vcvv 3496  ∅c0 4293  {csn 4569  ⟨cop 4575  ∪ ciun 4921   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  dom cdm 5557  ran crn 5558   ↾ cres 5559  suc csuc 6195  –1-1→wf1 6354  –onto→wfo 6355  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  ωcom 7582  seq_ωcseqom 8085   ↑_m cmap 8408   ≈ cen 8508   ≼ cdom 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-seqom 8086  df-1o 8104  df-map 8410  df-en 8512  df-dom 8513

This theorem is referenced by:  infpwfien  9490