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Theorem fseqen 9050
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 8118 . 2 ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
2 n0 4078 . 2 (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏𝐴)
3 eeanv 2344 . . 3 (∃𝑓𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) ↔ (∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴 ∧ ∃𝑏 𝑏𝐴))
4 omex 8704 . . . . . . 7 ω ∈ V
5 simpl 468 . . . . . . . . 9 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
6 f1ofo 6285 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑓:(𝐴 × 𝐴)–onto𝐴)
7 forn 6259 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–onto𝐴 → ran 𝑓 = 𝐴)
85, 6, 73syl 18 . . . . . . . 8 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → ran 𝑓 = 𝐴)
9 vex 3354 . . . . . . . . 9 𝑓 ∈ V
109rnex 7247 . . . . . . . 8 ran 𝑓 ∈ V
118, 10syl6eqelr 2859 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝐴 ∈ V)
12 xpexg 7107 . . . . . . 7 ((ω ∈ V ∧ 𝐴 ∈ V) → (ω × 𝐴) ∈ V)
134, 11, 12sylancr 575 . . . . . 6 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → (ω × 𝐴) ∈ V)
14 simpr 471 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑏𝐴)
15 eqid 2771 . . . . . . 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 2771 . . . . . . 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 9048 . . . . . 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 8129 . . . . . 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 9049 . . . . . 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 8236 . . . . 5 (( 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛)) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
2319, 21, 22syl2anc 573 . . . 4 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
2423exlimivv 2012 . . 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 585 1 (((𝐴 × 𝐴) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  Vcvv 3351  c0 4063  {csn 4316  cop 4322   ciun 4654   class class class wbr 4786  cmpt 4863   × cxp 5247  dom cdm 5249  ran crn 5250  cres 5251  suc csuc 5868  1-1wf1 6028  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  cmpt2 6795  ωcom 7212  seq𝜔cseqom 7695  𝑚 cmap 8009  cen 8106  cdom 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-1o 7713  df-map 8011  df-en 8110  df-dom 8111
This theorem is referenced by:  infpwfien  9085
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