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Theorem fseqen 8795
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) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴𝑚 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.
StepHypRef Expression
1 bren 7909 . 2 ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
2 n0 3912 . 2 (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏𝐴)
3 eeanv 2186 . . 3 (∃𝑓𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) ↔ (∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴 ∧ ∃𝑏 𝑏𝐴))
4 omex 8485 . . . . . . 7 ω ∈ V
5 simpl 473 . . . . . . . . 9 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
6 f1ofo 6103 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑓:(𝐴 × 𝐴)–onto𝐴)
7 forn 6077 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–onto𝐴 → ran 𝑓 = 𝐴)
85, 6, 73syl 18 . . . . . . . 8 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → ran 𝑓 = 𝐴)
9 vex 3194 . . . . . . . . 9 𝑓 ∈ V
109rnex 7048 . . . . . . . 8 ran 𝑓 ∈ V
118, 10syl6eqelr 2713 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝐴 ∈ V)
12 xpexg 6914 . . . . . . 7 ((ω ∈ V ∧ 𝐴 ∈ V) → (ω × 𝐴) ∈ V)
134, 11, 12sylancr 694 . . . . . 6 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → (ω × 𝐴) ∈ V)
14 simpr 477 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑏𝐴)
15 eqid 2626 . . . . . . 7 seq𝜔((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴𝑚 suc 𝑘) ↦ ((𝑔‘(𝑦𝑘))𝑓(𝑦𝑘)))), {⟨∅, 𝑏⟩}) = seq𝜔((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴𝑚 suc 𝑘) ↦ ((𝑔‘(𝑦𝑘))𝑓(𝑦𝑘)))), {⟨∅, 𝑏⟩})
16 eqid 2626 . . . . . . 7 (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ⟨dom 𝑥, ((seq𝜔((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴𝑚 suc 𝑘) ↦ ((𝑔‘(𝑦𝑘))𝑓(𝑦𝑘)))), {⟨∅, 𝑏⟩})‘dom 𝑥)‘𝑥)⟩) = (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ⟨dom 𝑥, ((seq𝜔((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴𝑚 suc 𝑘) ↦ ((𝑔‘(𝑦𝑘))𝑓(𝑦𝑘)))), {⟨∅, 𝑏⟩})‘dom 𝑥)‘𝑥)⟩)
1711, 14, 5, 15, 16fseqenlem2 8793 . . . . . 6 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ⟨dom 𝑥, ((seq𝜔((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴𝑚 suc 𝑘) ↦ ((𝑔‘(𝑦𝑘))𝑓(𝑦𝑘)))), {⟨∅, 𝑏⟩})‘dom 𝑥)‘𝑥)⟩): 𝑛 ∈ ω (𝐴𝑚 𝑛)–1-1→(ω × 𝐴))
18 f1domg 7920 . . . . . 6 ((ω × 𝐴) ∈ V → ((𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ⟨dom 𝑥, ((seq𝜔((𝑘 ∈ V, 𝑔 ∈ V ↦ (𝑦 ∈ (𝐴𝑚 suc 𝑘) ↦ ((𝑔‘(𝑦𝑘))𝑓(𝑦𝑘)))), {⟨∅, 𝑏⟩})‘dom 𝑥)‘𝑥)⟩): 𝑛 ∈ ω (𝐴𝑚 𝑛)–1-1→(ω × 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ (ω × 𝐴)))
1913, 17, 18sylc 65 . . . . 5 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ (ω × 𝐴))
20 fseqdom 8794 . . . . . 6 (𝐴 ∈ V → (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛))
2111, 20syl 17 . . . . 5 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛))
22 sbth 8025 . . . . 5 (( 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛)) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
2319, 21, 22syl2anc 692 . . . 4 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
2423exlimivv 1862 . . 3 (∃𝑓𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
253, 24sylbir 225 . 2 ((∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴 ∧ ∃𝑏 𝑏𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
261, 2, 25syl2anb 496 1 (((𝐴 × 𝐴) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1992  wne 2796  Vcvv 3191  c0 3896  {csn 4153  cop 4159   ciun 4490   class class class wbr 4618  cmpt 4678   × cxp 5077  dom cdm 5079  ran crn 5080  cres 5081  suc csuc 5687  1-1wf1 5847  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  cmpt2 6607  ωcom 7013  seq𝜔cseqom 7488  𝑚 cmap 7803  cen 7897  cdom 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-seqom 7489  df-1o 7506  df-map 7805  df-en 7901  df-dom 7902
This theorem is referenced by:  infpwfien  8830
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