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Theorem fseqen 9440
 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 8503 . 2 ((𝐴 × 𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
2 n0 4260 . 2 (𝐴 ≠ ∅ ↔ ∃𝑏 𝑏𝐴)
3 exdistrv 1956 . . 3 (∃𝑓𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) ↔ (∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴 ∧ ∃𝑏 𝑏𝐴))
4 omex 9092 . . . . . . 7 ω ∈ V
5 simpl 486 . . . . . . . . 9 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴)
6 f1ofo 6597 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑓:(𝐴 × 𝐴)–onto𝐴)
7 forn 6568 . . . . . . . . 9 (𝑓:(𝐴 × 𝐴)–onto𝐴 → ran 𝑓 = 𝐴)
85, 6, 73syl 18 . . . . . . . 8 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → ran 𝑓 = 𝐴)
9 vex 3444 . . . . . . . . 9 𝑓 ∈ V
109rnex 7601 . . . . . . . 8 ran 𝑓 ∈ V
118, 10eqeltrrdi 2899 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝐴 ∈ V)
12 xpexg 7455 . . . . . . 7 ((ω ∈ V ∧ 𝐴 ∈ V) → (ω × 𝐴) ∈ V)
134, 11, 12sylancr 590 . . . . . 6 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → (ω × 𝐴) ∈ V)
14 simpr 488 . . . . . . 7 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑏𝐴)
15 eqid 2798 . . . . . . 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 2798 . . . . . . 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 9438 . . . . . 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 8514 . . . . . 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 9439 . . . . . 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 8623 . . . . 5 (( 𝑛 ∈ ω (𝐴m 𝑛) ≼ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴m 𝑛)) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
2319, 21, 22syl2anc 587 . . . 4 ((𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
2423exlimivv 1933 . . 3 (∃𝑓𝑏(𝑓:(𝐴 × 𝐴)–1-1-onto𝐴𝑏𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
253, 24sylbir 238 . 2 ((∃𝑓 𝑓:(𝐴 × 𝐴)–1-1-onto𝐴 ∧ ∃𝑏 𝑏𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
261, 2, 25syl2anb 600 1 (((𝐴 × 𝐴) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441  ∅c0 4243  {csn 4525  ⟨cop 4531  ∪ ciun 4881   class class class wbr 5030   ↦ cmpt 5110   × cxp 5517  dom cdm 5519  ran crn 5520   ↾ cres 5521  suc csuc 6161  –1-1→wf1 6321  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  ωcom 7562  seqωcseqom 8068   ↑m cmap 8391   ≈ cen 8491   ≼ cdom 8492 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-inf2 9090 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-seqom 8069  df-1o 8087  df-map 8393  df-en 8495  df-dom 8496 This theorem is referenced by:  infpwfien  9475
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