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Theorem fseqen 9060
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 8132 . 2 ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
2 n0 4074 . 2 (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏𝐴)
3 eeanv 2327 . . 3 (∃𝑓𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) ↔ (∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴 ∧ ∃𝑏 𝑏𝐴))
4 omex 8715 . . . . . . 7 ω ∈ V
5 simpl 474 . . . . . . . . 9 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
6 f1ofo 6306 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑓:(𝐴 × 𝐴)–onto𝐴)
7 forn 6280 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–onto𝐴 → ran 𝑓 = 𝐴)
85, 6, 73syl 18 . . . . . . . 8 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → ran 𝑓 = 𝐴)
9 vex 3343 . . . . . . . . 9 𝑓 ∈ V
109rnex 7266 . . . . . . . 8 ran 𝑓 ∈ V
118, 10syl6eqelr 2848 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝐴 ∈ V)
12 xpexg 7126 . . . . . . 7 ((ω ∈ V ∧ 𝐴 ∈ V) → (ω × 𝐴) ∈ V)
134, 11, 12sylancr 698 . . . . . 6 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → (ω × 𝐴) ∈ V)
14 simpr 479 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑏𝐴)
15 eqid 2760 . . . . . . 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 2760 . . . . . . 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 9058 . . . . . 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 8143 . . . . . 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 9059 . . . . . 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 8247 . . . . 5 (( 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛)) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
2319, 21, 22syl2anc 696 . . . 4 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
2423exlimivv 2009 . . 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 497 1 (((𝐴 × 𝐴) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wex 1853  wcel 2139  wne 2932  Vcvv 3340  c0 4058  {csn 4321  cop 4327   ciun 4672   class class class wbr 4804  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267  cres 5268  suc csuc 5886  1-1wf1 6046  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6814  cmpt2 6816  ωcom 7231  seq𝜔cseqom 7712  𝑚 cmap 8025  cen 8120  cdom 8121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-seqom 7713  df-1o 7730  df-map 8027  df-en 8124  df-dom 8125
This theorem is referenced by:  infpwfien  9095
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