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Theorem fseqen 9968
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 8896 . 2 ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
2 n0 4307 . 2 (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏𝐴)
3 exdistrv 1960 . . 3 (∃𝑓𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) ↔ (∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴 ∧ ∃𝑏 𝑏𝐴))
4 omex 9584 . . . . . . 7 ω ∈ V
5 simpl 484 . . . . . . . . 9 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
6 f1ofo 6792 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑓:(𝐴 × 𝐴)–onto𝐴)
7 forn 6760 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–onto𝐴 → ran 𝑓 = 𝐴)
85, 6, 73syl 18 . . . . . . . 8 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → ran 𝑓 = 𝐴)
9 vex 3448 . . . . . . . . 9 𝑓 ∈ V
109rnex 7850 . . . . . . . 8 ran 𝑓 ∈ V
118, 10eqeltrrdi 2843 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝐴 ∈ V)
12 xpexg 7685 . . . . . . 7 ((ω ∈ V ∧ 𝐴 ∈ V) → (ω × 𝐴) ∈ V)
134, 11, 12sylancr 588 . . . . . 6 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → (ω × 𝐴) ∈ V)
14 simpr 486 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑏𝐴)
15 eqid 2733 . . . . . . 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 2733 . . . . . . 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 9966 . . . . . 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 8915 . . . . . 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 9967 . . . . . 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 9040 . . . . 5 (( 𝑛 ∈ ω (𝐴m 𝑛) ≼ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴m 𝑛)) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
2319, 21, 22syl2anc 585 . . . 4 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
2423exlimivv 1936 . . 3 (∃𝑓𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
253, 24sylbir 234 . 2 ((∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴 ∧ ∃𝑏 𝑏𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
261, 2, 25syl2anb 599 1 (((𝐴 × 𝐴) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2940  Vcvv 3444  c0 4283  {csn 4587  cop 4593   ciun 4955   class class class wbr 5106  cmpt 5189   × cxp 5632  dom cdm 5634  ran crn 5635  cres 5636  suc csuc 6320  1-1wf1 6494  ontowfo 6495  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  cmpo 7360  ωcom 7803  seqωcseqom 8394  m cmap 8768  cen 8883  cdom 8884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-seqom 8395  df-1o 8413  df-map 8770  df-en 8887  df-dom 8888
This theorem is referenced by:  infpwfien  10003
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