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Theorem iundisj 25297
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 4076 . . . . . . . . . 10 {𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ ℕ
2 nnuz 12869 . . . . . . . . . 10 ℕ = (ℤ‘1)
31, 2sseqtri 4017 . . . . . . . . 9 {𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ (ℤ‘1)
4 rabn0 4384 . . . . . . . . . 10 ({𝑛 ∈ ℕ ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑛 ∈ ℕ 𝑥𝐴)
54biimpri 227 . . . . . . . . 9 (∃𝑛 ∈ ℕ 𝑥𝐴 → {𝑛 ∈ ℕ ∣ 𝑥𝐴} ≠ ∅)
6 infssuzcl 12920 . . . . . . . . 9 (({𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ {𝑛 ∈ ℕ ∣ 𝑥𝐴} ≠ ∅) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴})
73, 5, 6sylancr 585 . . . . . . . 8 (∃𝑛 ∈ ℕ 𝑥𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴})
8 nfrab1 3449 . . . . . . . . . 10 𝑛{𝑛 ∈ ℕ ∣ 𝑥𝐴}
9 nfcv 2901 . . . . . . . . . 10 𝑛
10 nfcv 2901 . . . . . . . . . 10 𝑛 <
118, 9, 10nfinf 9479 . . . . . . . . 9 𝑛inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )
12 nfcv 2901 . . . . . . . . 9 𝑛
1311nfcsb1 3916 . . . . . . . . . 10 𝑛inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
1413nfcri 2888 . . . . . . . . 9 𝑛 𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
15 csbeq1a 3906 . . . . . . . . . 10 (𝑛 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → 𝐴 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
1615eleq2d 2817 . . . . . . . . 9 (𝑛 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → (𝑥𝐴𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
1711, 12, 14, 16elrabf 3678 . . . . . . . 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 493 . . . . . 6 (∃𝑛 ∈ ℕ 𝑥𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℕ)
2018simprd 494 . . . . . . 7 (∃𝑛 ∈ ℕ 𝑥𝐴𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
2119nnred 12231 . . . . . . . . 9 (∃𝑛 ∈ ℕ 𝑥𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
2221ltnrd 11352 . . . . . . . 8 (∃𝑛 ∈ ℕ 𝑥𝐴 → ¬ inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
23 eliun 5000 . . . . . . . . 9 (𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵 ↔ ∃𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝑥𝐵)
2421ad2antrr 722 . . . . . . . . . . 11 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
25 elfzouz 13640 . . . . . . . . . . . . . 14 (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 ∈ (ℤ‘1))
2625, 2eleqtrrdi 2842 . . . . . . . . . . . . 13 (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 ∈ ℕ)
2726ad2antlr 723 . . . . . . . . . . . 12 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℕ)
2827nnred 12231 . . . . . . . . . . 11 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℝ)
29 iundisj.1 . . . . . . . . . . . . . 14 (𝑛 = 𝑘𝐴 = 𝐵)
3029eleq2d 2817 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑥𝐴𝑥𝐵))
31 simpr 483 . . . . . . . . . . . . 13 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑥𝐵)
3230, 27, 31elrabd 3684 . . . . . . . . . . . 12 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴})
33 infssuzle 12919 . . . . . . . . . . . 12 (({𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴}) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
343, 32, 33sylancr 585 . . . . . . . . . . 11 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
35 elfzolt2 13645 . . . . . . . . . . . 12 (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
3635ad2antlr 723 . . . . . . . . . . 11 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
3724, 28, 24, 34, 36lelttrd 11376 . . . . . . . . . 10 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
3837rexlimdva2 3155 . . . . . . . . 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 3958 . . . . . 6 (∃𝑛 ∈ ℕ 𝑥𝐴𝑥 ∈ (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵))
42 csbeq1 3895 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → 𝑚 / 𝑛𝐴 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
43 oveq2 7419 . . . . . . . . . 10 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → (1..^𝑚) = (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )))
4443iuneq1d 5023 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → 𝑘 ∈ (1..^𝑚)𝐵 = 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵)
4542, 44difeq12d 4122 . . . . . . . 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 2817 . . . . . . 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 3611 . . . . . 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 582 . . . . 5 (∃𝑛 ∈ ℕ 𝑥𝐴 → ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
49 nfv 1915 . . . . . 6 𝑚 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
50 nfcsb1v 3917 . . . . . . . 8 𝑛𝑚 / 𝑛𝐴
51 nfcv 2901 . . . . . . . 8 𝑛 𝑘 ∈ (1..^𝑚)𝐵
5250, 51nfdif 4124 . . . . . . 7 𝑛(𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
5352nfcri 2888 . . . . . 6 𝑛 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
54 csbeq1a 3906 . . . . . . . 8 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
55 oveq2 7419 . . . . . . . . 9 (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
5655iuneq1d 5023 . . . . . . . 8 (𝑛 = 𝑚 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^𝑚)𝐵)
5754, 56difeq12d 4122 . . . . . . 7 (𝑛 = 𝑚 → (𝐴 𝑘 ∈ (1..^𝑛)𝐵) = (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
5857eleq2d 2817 . . . . . 6 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)))
5949, 53, 58cbvrexw 3302 . . . . 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 4125 . . . . 5 (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → 𝑥𝐴)
6261reximi 3082 . . . 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 5000 . . 3 (𝑥 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑥𝐴)
65 eliun 5000 . . 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 2727 1 𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wne 2938  wrex 3068  {crab 3430  csb 3892  cdif 3944  wss 3947  c0 4321   ciun 4996   class class class wbr 5147  cfv 6542  (class class class)co 7411  infcinf 9438  cr 11111  1c1 11113   < clt 11252  cle 11253  cn 12216  cuz 12826  ..^cfzo 13631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632
This theorem is referenced by:  iunmbl  25302  volsup  25305  sigapildsys  33458  carsgclctunlem3  33617  voliunnfl  36835
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