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Theorem iundisj 24896
Description: Rewrite a countable union as a disjoint union. (Contributed by Mario Carneiro, 20-Mar-2014.)
Hypothesis
Ref Expression
iundisj.1 (𝑛 = 𝑘𝐴 = 𝐵)
Assertion
Ref Expression
iundisj 𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
Distinct variable groups:   𝑘,𝑛   𝐴,𝑘   𝐵,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝐵(𝑘)

Proof of Theorem iundisj
Dummy variables 𝑥 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 4035 . . . . . . . . . 10 {𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ ℕ
2 nnuz 12798 . . . . . . . . . 10 ℕ = (ℤ‘1)
31, 2sseqtri 3978 . . . . . . . . 9 {𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ (ℤ‘1)
4 rabn0 4343 . . . . . . . . . 10 ({𝑛 ∈ ℕ ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑛 ∈ ℕ 𝑥𝐴)
54biimpri 227 . . . . . . . . 9 (∃𝑛 ∈ ℕ 𝑥𝐴 → {𝑛 ∈ ℕ ∣ 𝑥𝐴} ≠ ∅)
6 infssuzcl 12849 . . . . . . . . 9 (({𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ {𝑛 ∈ ℕ ∣ 𝑥𝐴} ≠ ∅) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴})
73, 5, 6sylancr 587 . . . . . . . 8 (∃𝑛 ∈ ℕ 𝑥𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴})
8 nfrab1 3424 . . . . . . . . . 10 𝑛{𝑛 ∈ ℕ ∣ 𝑥𝐴}
9 nfcv 2905 . . . . . . . . . 10 𝑛
10 nfcv 2905 . . . . . . . . . 10 𝑛 <
118, 9, 10nfinf 9414 . . . . . . . . 9 𝑛inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )
12 nfcv 2905 . . . . . . . . 9 𝑛
1311nfcsb1 3877 . . . . . . . . . 10 𝑛inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
1413nfcri 2892 . . . . . . . . 9 𝑛 𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
15 csbeq1a 3867 . . . . . . . . . 10 (𝑛 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → 𝐴 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
1615eleq2d 2823 . . . . . . . . 9 (𝑛 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → (𝑥𝐴𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
1711, 12, 14, 16elrabf 3639 . . . . . . . 8 (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴} ↔ (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℕ ∧ 𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
187, 17sylib 217 . . . . . . 7 (∃𝑛 ∈ ℕ 𝑥𝐴 → (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℕ ∧ 𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
1918simpld 495 . . . . . 6 (∃𝑛 ∈ ℕ 𝑥𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℕ)
2018simprd 496 . . . . . . 7 (∃𝑛 ∈ ℕ 𝑥𝐴𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
2119nnred 12164 . . . . . . . . 9 (∃𝑛 ∈ ℕ 𝑥𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
2221ltnrd 11285 . . . . . . . 8 (∃𝑛 ∈ ℕ 𝑥𝐴 → ¬ inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
23 eliun 4956 . . . . . . . . 9 (𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵 ↔ ∃𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝑥𝐵)
2421ad2antrr 724 . . . . . . . . . . 11 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
25 elfzouz 13568 . . . . . . . . . . . . . 14 (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 ∈ (ℤ‘1))
2625, 2eleqtrrdi 2849 . . . . . . . . . . . . 13 (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 ∈ ℕ)
2726ad2antlr 725 . . . . . . . . . . . 12 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℕ)
2827nnred 12164 . . . . . . . . . . 11 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℝ)
29 iundisj.1 . . . . . . . . . . . . . 14 (𝑛 = 𝑘𝐴 = 𝐵)
3029eleq2d 2823 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑥𝐴𝑥𝐵))
31 simpr 485 . . . . . . . . . . . . 13 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑥𝐵)
3230, 27, 31elrabd 3645 . . . . . . . . . . . 12 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴})
33 infssuzle 12848 . . . . . . . . . . . 12 (({𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴}) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
343, 32, 33sylancr 587 . . . . . . . . . . 11 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
35 elfzolt2 13573 . . . . . . . . . . . 12 (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
3635ad2antlr 725 . . . . . . . . . . 11 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
3724, 28, 24, 34, 36lelttrd 11309 . . . . . . . . . 10 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
3837rexlimdva2 3152 . . . . . . . . 9 (∃𝑛 ∈ ℕ 𝑥𝐴 → (∃𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝑥𝐵 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )))
3923, 38biimtrid 241 . . . . . . . 8 (∃𝑛 ∈ ℕ 𝑥𝐴 → (𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )))
4022, 39mtod 197 . . . . . . 7 (∃𝑛 ∈ ℕ 𝑥𝐴 → ¬ 𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵)
4120, 40eldifd 3919 . . . . . 6 (∃𝑛 ∈ ℕ 𝑥𝐴𝑥 ∈ (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵))
42 csbeq1 3856 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → 𝑚 / 𝑛𝐴 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
43 oveq2 7361 . . . . . . . . . 10 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → (1..^𝑚) = (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )))
4443iuneq1d 4979 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → 𝑘 ∈ (1..^𝑚)𝐵 = 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵)
4542, 44difeq12d 4081 . . . . . . . 8 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵) = (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵))
4645eleq2d 2823 . . . . . . 7 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → (𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵) ↔ 𝑥 ∈ (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵)))
4746rspcev 3579 . . . . . 6 ((inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℕ ∧ 𝑥 ∈ (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵)) → ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
4819, 41, 47syl2anc 584 . . . . 5 (∃𝑛 ∈ ℕ 𝑥𝐴 → ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
49 nfv 1917 . . . . . 6 𝑚 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
50 nfcsb1v 3878 . . . . . . . 8 𝑛𝑚 / 𝑛𝐴
51 nfcv 2905 . . . . . . . 8 𝑛 𝑘 ∈ (1..^𝑚)𝐵
5250, 51nfdif 4083 . . . . . . 7 𝑛(𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
5352nfcri 2892 . . . . . 6 𝑛 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
54 csbeq1a 3867 . . . . . . . 8 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
55 oveq2 7361 . . . . . . . . 9 (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
5655iuneq1d 4979 . . . . . . . 8 (𝑛 = 𝑚 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^𝑚)𝐵)
5754, 56difeq12d 4081 . . . . . . 7 (𝑛 = 𝑚 → (𝐴 𝑘 ∈ (1..^𝑛)𝐵) = (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
5857eleq2d 2823 . . . . . 6 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)))
5949, 53, 58cbvrexw 3288 . . . . 5 (∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
6048, 59sylibr 233 . . . 4 (∃𝑛 ∈ ℕ 𝑥𝐴 → ∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
61 eldifi 4084 . . . . 5 (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → 𝑥𝐴)
6261reximi 3085 . . . 4 (∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → ∃𝑛 ∈ ℕ 𝑥𝐴)
6360, 62impbii 208 . . 3 (∃𝑛 ∈ ℕ 𝑥𝐴 ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
64 eliun 4956 . . 3 (𝑥 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑥𝐴)
65 eliun 4956 . . 3 (𝑥 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
6663, 64, 653bitr4i 302 . 2 (𝑥 𝑛 ∈ ℕ 𝐴𝑥 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
6766eqriv 2733 1 𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2941  wrex 3071  {crab 3405  csb 3853  cdif 3905  wss 3908  c0 4280   ciun 4952   class class class wbr 5103  cfv 6493  (class class class)co 7353  infcinf 9373  cr 11046  1c1 11048   < clt 11185  cle 11186  cn 12149  cuz 12759  ..^cfzo 13559
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-er 8644  df-en 8880  df-dom 8881  df-sdom 8882  df-sup 9374  df-inf 9375  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-n0 12410  df-z 12496  df-uz 12760  df-fz 13417  df-fzo 13560
This theorem is referenced by:  iunmbl  24901  volsup  24904  sigapildsys  32630  carsgclctunlem3  32789  voliunnfl  36089
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