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Theorem iuninc 30896
Description: The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Hypotheses
Ref Expression
iuninc.1 (𝜑𝐹 Fn ℕ)
iuninc.2 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
Assertion
Ref Expression
iuninc ((𝜑𝑖 ∈ ℕ) → 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))
Distinct variable groups:   𝑖,𝑛   𝑛,𝐹   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑖)   𝐹(𝑖)

Proof of Theorem iuninc
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7279 . . . . . 6 (𝑗 = 1 → (1...𝑗) = (1...1))
21iuneq1d 4957 . . . . 5 (𝑗 = 1 → 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...1)(𝐹𝑛))
3 fveq2 6771 . . . . 5 (𝑗 = 1 → (𝐹𝑗) = (𝐹‘1))
42, 3eqeq12d 2756 . . . 4 (𝑗 = 1 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)))
54imbi2d 341 . . 3 (𝑗 = 1 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1))))
6 oveq2 7279 . . . . . 6 (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
76iuneq1d 4957 . . . . 5 (𝑗 = 𝑘 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
8 fveq2 6771 . . . . 5 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
97, 8eqeq12d 2756 . . . 4 (𝑗 = 𝑘 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)))
109imbi2d 341 . . 3 (𝑗 = 𝑘 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘))))
11 oveq2 7279 . . . . . 6 (𝑗 = (𝑘 + 1) → (1...𝑗) = (1...(𝑘 + 1)))
1211iuneq1d 4957 . . . . 5 (𝑗 = (𝑘 + 1) → 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
13 fveq2 6771 . . . . 5 (𝑗 = (𝑘 + 1) → (𝐹𝑗) = (𝐹‘(𝑘 + 1)))
1412, 13eqeq12d 2756 . . . 4 (𝑗 = (𝑘 + 1) → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1))))
1514imbi2d 341 . . 3 (𝑗 = (𝑘 + 1) → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
16 oveq2 7279 . . . . . 6 (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
1716iuneq1d 4957 . . . . 5 (𝑗 = 𝑖 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...𝑖)(𝐹𝑛))
18 fveq2 6771 . . . . 5 (𝑗 = 𝑖 → (𝐹𝑗) = (𝐹𝑖))
1917, 18eqeq12d 2756 . . . 4 (𝑗 = 𝑖 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖)))
2019imbi2d 341 . . 3 (𝑗 = 𝑖 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))))
21 1z 12350 . . . . . 6 1 ∈ ℤ
22 fzsn 13297 . . . . . 6 (1 ∈ ℤ → (1...1) = {1})
23 iuneq1 4946 . . . . . 6 ((1...1) = {1} → 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛))
2421, 22, 23mp2b 10 . . . . 5 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛)
25 1ex 10972 . . . . . 6 1 ∈ V
26 fveq2 6771 . . . . . 6 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
2725, 26iunxsn 5025 . . . . 5 𝑛 ∈ {1} (𝐹𝑛) = (𝐹‘1)
2824, 27eqtri 2768 . . . 4 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)
2928a1i 11 . . 3 (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1))
30 simpll 764 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑘 ∈ ℕ)
31 elnnuz 12621 . . . . . . . . . 10 (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ‘1))
32 fzsuc 13302 . . . . . . . . . 10 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
3331, 32sylbi 216 . . . . . . . . 9 (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
3433iuneq1d 4957 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
35 iunxun 5028 . . . . . . . . 9 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛))
36 ovex 7304 . . . . . . . . . . 11 (𝑘 + 1) ∈ V
37 fveq2 6771 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
3836, 37iunxsn 5025 . . . . . . . . . 10 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛) = (𝐹‘(𝑘 + 1))
3938uneq2i 4099 . . . . . . . . 9 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛)) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
4035, 39eqtri 2768 . . . . . . . 8 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
4134, 40eqtrdi 2796 . . . . . . 7 (𝑘 ∈ ℕ → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))))
4230, 41syl 17 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))))
43 simpr 485 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘))
4443uneq1d 4101 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))))
45 simplr 766 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝜑)
46 iuninc.2 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
4746sbt 2073 . . . . . . . . 9 [𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
48 sbim 2304 . . . . . . . . . 10 ([𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))))
49 sban 2087 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ))
50 sbv 2095 . . . . . . . . . . . . 13 ([𝑘 / 𝑛]𝜑𝜑)
51 clelsb1 2868 . . . . . . . . . . . . 13 ([𝑘 / 𝑛]𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)
5250, 51anbi12i 627 . . . . . . . . . . . 12 (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ) ↔ (𝜑𝑘 ∈ ℕ))
5349, 52bitr2i 275 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) ↔ [𝑘 / 𝑛](𝜑𝑛 ∈ ℕ))
54 sbsbc 3724 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
55 sbcssg 4460 . . . . . . . . . . . . 13 (𝑘 ∈ V → ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ 𝑘 / 𝑛(𝐹𝑛) ⊆ 𝑘 / 𝑛(𝐹‘(𝑛 + 1))))
5655elv 3437 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ 𝑘 / 𝑛(𝐹𝑛) ⊆ 𝑘 / 𝑛(𝐹‘(𝑛 + 1)))
57 csbfv 6816 . . . . . . . . . . . . 13 𝑘 / 𝑛(𝐹𝑛) = (𝐹𝑘)
58 csbfv2g 6815 . . . . . . . . . . . . . . 15 (𝑘 ∈ V → 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹𝑘 / 𝑛(𝑛 + 1)))
5958elv 3437 . . . . . . . . . . . . . 14 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹𝑘 / 𝑛(𝑛 + 1))
60 csbov1g 7316 . . . . . . . . . . . . . . . 16 (𝑘 ∈ V → 𝑘 / 𝑛(𝑛 + 1) = (𝑘 / 𝑛𝑛 + 1))
6160elv 3437 . . . . . . . . . . . . . . 15 𝑘 / 𝑛(𝑛 + 1) = (𝑘 / 𝑛𝑛 + 1)
6261fveq2i 6774 . . . . . . . . . . . . . 14 (𝐹𝑘 / 𝑛(𝑛 + 1)) = (𝐹‘(𝑘 / 𝑛𝑛 + 1))
63 vex 3435 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
6463csbvargi 4372 . . . . . . . . . . . . . . . 16 𝑘 / 𝑛𝑛 = 𝑘
6564oveq1i 7281 . . . . . . . . . . . . . . 15 (𝑘 / 𝑛𝑛 + 1) = (𝑘 + 1)
6665fveq2i 6774 . . . . . . . . . . . . . 14 (𝐹‘(𝑘 / 𝑛𝑛 + 1)) = (𝐹‘(𝑘 + 1))
6759, 62, 663eqtri 2772 . . . . . . . . . . . . 13 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))
6857, 67sseq12i 3956 . . . . . . . . . . . 12 (𝑘 / 𝑛(𝐹𝑛) ⊆ 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) ↔ (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)))
6954, 56, 683bitrri 298 . . . . . . . . . . 11 ((𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
7053, 69imbi12i 351 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1))) ↔ ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))))
7148, 70bitr4i 277 . . . . . . . . 9 ([𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1))))
7247, 71mpbi 229 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)))
73 ssequn1 4119 . . . . . . . 8 ((𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7472, 73sylib 217 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7545, 30, 74syl2anc 584 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7642, 44, 753eqtrd 2784 . . . . 5 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))
7776exp31 420 . . . 4 (𝑘 ∈ ℕ → (𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
7877a2d 29 . . 3 (𝑘 ∈ ℕ → ((𝜑 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → (𝜑 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
795, 10, 15, 20, 29, 78nnind 11991 . 2 (𝑖 ∈ ℕ → (𝜑 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖)))
8079impcom 408 1 ((𝜑𝑖 ∈ ℕ) → 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1542  [wsb 2071  wcel 2110  Vcvv 3431  [wsbc 3720  csb 3837  cun 3890  wss 3892  {csn 4567   ciun 4930   Fn wfn 6427  cfv 6432  (class class class)co 7271  1c1 10873   + caddc 10875  cn 11973  cz 12319  cuz 12581  ...cfz 13238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12582  df-fz 13239
This theorem is referenced by:  meascnbl  32183
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