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Theorem iundisj 23239
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 3671 . . . . . . . . . 10 {𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ ℕ
2 nnuz 11675 . . . . . . . . . 10 ℕ = (ℤ‘1)
31, 2sseqtri 3621 . . . . . . . . 9 {𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ (ℤ‘1)
4 rabn0 3937 . . . . . . . . . 10 ({𝑛 ∈ ℕ ∣ 𝑥𝐴} ≠ ∅ ↔ ∃𝑛 ∈ ℕ 𝑥𝐴)
54biimpri 218 . . . . . . . . 9 (∃𝑛 ∈ ℕ 𝑥𝐴 → {𝑛 ∈ ℕ ∣ 𝑥𝐴} ≠ ∅)
6 infssuzcl 11724 . . . . . . . . 9 (({𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ {𝑛 ∈ ℕ ∣ 𝑥𝐴} ≠ ∅) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴})
73, 5, 6sylancr 694 . . . . . . . 8 (∃𝑛 ∈ ℕ 𝑥𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴})
8 nfrab1 3114 . . . . . . . . . 10 𝑛{𝑛 ∈ ℕ ∣ 𝑥𝐴}
9 nfcv 2761 . . . . . . . . . 10 𝑛
10 nfcv 2761 . . . . . . . . . 10 𝑛 <
118, 9, 10nfinf 8340 . . . . . . . . 9 𝑛inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )
12 nfcv 2761 . . . . . . . . 9 𝑛
1311nfcsb1 3533 . . . . . . . . . 10 𝑛inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
1413nfcri 2755 . . . . . . . . 9 𝑛 𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴
15 csbeq1a 3527 . . . . . . . . . 10 (𝑛 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → 𝐴 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
1615eleq2d 2684 . . . . . . . . 9 (𝑛 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → (𝑥𝐴𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
1711, 12, 14, 16elrabf 3347 . . . . . . . 8 (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴} ↔ (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℕ ∧ 𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
187, 17sylib 208 . . . . . . 7 (∃𝑛 ∈ ℕ 𝑥𝐴 → (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℕ ∧ 𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴))
1918simpld 475 . . . . . 6 (∃𝑛 ∈ ℕ 𝑥𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℕ)
2018simprd 479 . . . . . . 7 (∃𝑛 ∈ ℕ 𝑥𝐴𝑥inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
2119nnred 10987 . . . . . . . . 9 (∃𝑛 ∈ ℕ 𝑥𝐴 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
2221ltnrd 10123 . . . . . . . 8 (∃𝑛 ∈ ℕ 𝑥𝐴 → ¬ inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
23 eliun 4495 . . . . . . . . 9 (𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵 ↔ ∃𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝑥𝐵)
2421ad2antrr 761 . . . . . . . . . . . 12 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ∈ ℝ)
25 elfzouz 12423 . . . . . . . . . . . . . . 15 (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 ∈ (ℤ‘1))
2625, 2syl6eleqr 2709 . . . . . . . . . . . . . 14 (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 ∈ ℕ)
2726ad2antlr 762 . . . . . . . . . . . . 13 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℕ)
2827nnred 10987 . . . . . . . . . . . 12 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ ℝ)
29 simpr 477 . . . . . . . . . . . . . 14 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑥𝐵)
30 iundisj.1 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘𝐴 = 𝐵)
3130eleq2d 2684 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝑥𝐴𝑥𝐵))
3231elrab 3350 . . . . . . . . . . . . . 14 (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴} ↔ (𝑘 ∈ ℕ ∧ 𝑥𝐵))
3327, 29, 32sylanbrc 697 . . . . . . . . . . . . 13 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴})
34 infssuzle 11723 . . . . . . . . . . . . 13 (({𝑛 ∈ ℕ ∣ 𝑥𝐴} ⊆ (ℤ‘1) ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑥𝐴}) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
353, 33, 34sylancr 694 . . . . . . . . . . . 12 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) ≤ 𝑘)
36 elfzolt2 12428 . . . . . . . . . . . . 13 (𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )) → 𝑘 < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
3736ad2antlr 762 . . . . . . . . . . . 12 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → 𝑘 < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
3824, 28, 24, 35, 37lelttrd 10147 . . . . . . . . . . 11 (((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) ∧ 𝑥𝐵) → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))
3938ex 450 . . . . . . . . . 10 ((∃𝑛 ∈ ℕ 𝑥𝐴𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))) → (𝑥𝐵 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )))
4039rexlimdva 3025 . . . . . . . . 9 (∃𝑛 ∈ ℕ 𝑥𝐴 → (∃𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝑥𝐵 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )))
4123, 40syl5bi 232 . . . . . . . 8 (∃𝑛 ∈ ℕ 𝑥𝐴 → (𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵 → inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) < inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )))
4222, 41mtod 189 . . . . . . 7 (∃𝑛 ∈ ℕ 𝑥𝐴 → ¬ 𝑥 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵)
4320, 42eldifd 3570 . . . . . 6 (∃𝑛 ∈ ℕ 𝑥𝐴𝑥 ∈ (inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵))
44 csbeq1 3521 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → 𝑚 / 𝑛𝐴 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) / 𝑛𝐴)
45 oveq2 6618 . . . . . . . . . 10 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → (1..^𝑚) = (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < )))
4645iuneq1d 4516 . . . . . . . . 9 (𝑚 = inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ) → 𝑘 ∈ (1..^𝑚)𝐵 = 𝑘 ∈ (1..^inf({𝑛 ∈ ℕ ∣ 𝑥𝐴}, ℝ, < ))𝐵)
4744, 46difeq12d 3712 . . . . . . . 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))
4847eleq2d 2684 . . . . . . 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)))
4948rspcev 3298 . . . . . 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))
5019, 43, 49syl2anc 692 . . . . 5 (∃𝑛 ∈ ℕ 𝑥𝐴 → ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
51 nfv 1840 . . . . . 6 𝑚 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
52 nfcsb1v 3534 . . . . . . . 8 𝑛𝑚 / 𝑛𝐴
53 nfcv 2761 . . . . . . . 8 𝑛 𝑘 ∈ (1..^𝑚)𝐵
5452, 53nfdif 3714 . . . . . . 7 𝑛(𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
5554nfcri 2755 . . . . . 6 𝑛 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)
56 csbeq1a 3527 . . . . . . . 8 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
57 oveq2 6618 . . . . . . . . 9 (𝑛 = 𝑚 → (1..^𝑛) = (1..^𝑚))
5857iuneq1d 4516 . . . . . . . 8 (𝑛 = 𝑚 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^𝑚)𝐵)
5956, 58difeq12d 3712 . . . . . . 7 (𝑛 = 𝑚 → (𝐴 𝑘 ∈ (1..^𝑛)𝐵) = (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
6059eleq2d 2684 . . . . . 6 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵)))
6151, 55, 60cbvrex 3159 . . . . 5 (∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ (𝑚 / 𝑛𝐴 𝑘 ∈ (1..^𝑚)𝐵))
6250, 61sylibr 224 . . . 4 (∃𝑛 ∈ ℕ 𝑥𝐴 → ∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
63 eldifi 3715 . . . . 5 (𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → 𝑥𝐴)
6463reximi 3006 . . . 4 (∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) → ∃𝑛 ∈ ℕ 𝑥𝐴)
6562, 64impbii 199 . . 3 (∃𝑛 ∈ ℕ 𝑥𝐴 ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
66 eliun 4495 . . 3 (𝑥 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑥𝐴)
67 eliun 4495 . . 3 (𝑥 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵) ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
6865, 66, 673bitr4i 292 . 2 (𝑥 𝑛 ∈ ℕ 𝐴𝑥 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵))
6968eqriv 2618 1 𝑛 ∈ ℕ 𝐴 = 𝑛 ∈ ℕ (𝐴 𝑘 ∈ (1..^𝑛)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  csb 3518  cdif 3556  wss 3559  c0 3896   ciun 4490   class class class wbr 4618  cfv 5852  (class class class)co 6610  infcinf 8299  cr 9887  1c1 9889   < clt 10026  cle 10027  cn 10972  cuz 11639  ..^cfzo 12414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415
This theorem is referenced by:  iunmbl  23244  volsup  23247  sigapildsys  30030  carsgclctunlem3  30187  voliunnfl  33120
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