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Theorem iuninc 32323
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 7422 . . . . . 6 (𝑗 = 1 → (1...𝑗) = (1...1))
21iuneq1d 5018 . . . . 5 (𝑗 = 1 → 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...1)(𝐹𝑛))
3 fveq2 6891 . . . . 5 (𝑗 = 1 → (𝐹𝑗) = (𝐹‘1))
42, 3eqeq12d 2743 . . . 4 (𝑗 = 1 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)))
54imbi2d 340 . . 3 (𝑗 = 1 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1))))
6 oveq2 7422 . . . . . 6 (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
76iuneq1d 5018 . . . . 5 (𝑗 = 𝑘 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
8 fveq2 6891 . . . . 5 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
97, 8eqeq12d 2743 . . . 4 (𝑗 = 𝑘 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)))
109imbi2d 340 . . 3 (𝑗 = 𝑘 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘))))
11 oveq2 7422 . . . . . 6 (𝑗 = (𝑘 + 1) → (1...𝑗) = (1...(𝑘 + 1)))
1211iuneq1d 5018 . . . . 5 (𝑗 = (𝑘 + 1) → 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
13 fveq2 6891 . . . . 5 (𝑗 = (𝑘 + 1) → (𝐹𝑗) = (𝐹‘(𝑘 + 1)))
1412, 13eqeq12d 2743 . . . 4 (𝑗 = (𝑘 + 1) → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1))))
1514imbi2d 340 . . 3 (𝑗 = (𝑘 + 1) → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
16 oveq2 7422 . . . . . 6 (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
1716iuneq1d 5018 . . . . 5 (𝑗 = 𝑖 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...𝑖)(𝐹𝑛))
18 fveq2 6891 . . . . 5 (𝑗 = 𝑖 → (𝐹𝑗) = (𝐹𝑖))
1917, 18eqeq12d 2743 . . . 4 (𝑗 = 𝑖 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖)))
2019imbi2d 340 . . 3 (𝑗 = 𝑖 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))))
21 1z 12608 . . . . . 6 1 ∈ ℤ
22 fzsn 13561 . . . . . 6 (1 ∈ ℤ → (1...1) = {1})
23 iuneq1 5007 . . . . . 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 11226 . . . . . 6 1 ∈ V
26 fveq2 6891 . . . . . 6 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
2725, 26iunxsn 5088 . . . . 5 𝑛 ∈ {1} (𝐹𝑛) = (𝐹‘1)
2824, 27eqtri 2755 . . . 4 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)
2928a1i 11 . . 3 (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1))
30 simpll 766 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑘 ∈ ℕ)
31 elnnuz 12882 . . . . . . . . . 10 (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ‘1))
32 fzsuc 13566 . . . . . . . . . 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 5018 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
35 iunxun 5091 . . . . . . . . 9 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛))
36 ovex 7447 . . . . . . . . . . 11 (𝑘 + 1) ∈ V
37 fveq2 6891 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
3836, 37iunxsn 5088 . . . . . . . . . 10 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛) = (𝐹‘(𝑘 + 1))
3938uneq2i 4156 . . . . . . . . 9 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛)) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
4035, 39eqtri 2755 . . . . . . . 8 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
4134, 40eqtrdi 2783 . . . . . . 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 484 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘))
4443uneq1d 4158 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))))
45 simplr 768 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝜑)
46 iuninc.2 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
4746sbt 2062 . . . . . . . . 9 [𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
48 sbim 2292 . . . . . . . . . 10 ([𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))))
49 sban 2076 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ))
50 sbv 2084 . . . . . . . . . . . . 13 ([𝑘 / 𝑛]𝜑𝜑)
51 clelsb1 2855 . . . . . . . . . . . . 13 ([𝑘 / 𝑛]𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)
5250, 51anbi12i 626 . . . . . . . . . . . 12 (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ) ↔ (𝜑𝑘 ∈ ℕ))
5349, 52bitr2i 276 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) ↔ [𝑘 / 𝑛](𝜑𝑛 ∈ ℕ))
54 sbsbc 3778 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
55 sbcssg 4519 . . . . . . . . . . . . 13 (𝑘 ∈ V → ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ 𝑘 / 𝑛(𝐹𝑛) ⊆ 𝑘 / 𝑛(𝐹‘(𝑛 + 1))))
5655elv 3475 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ 𝑘 / 𝑛(𝐹𝑛) ⊆ 𝑘 / 𝑛(𝐹‘(𝑛 + 1)))
57 csbfv 6941 . . . . . . . . . . . . 13 𝑘 / 𝑛(𝐹𝑛) = (𝐹𝑘)
58 csbfv2g 6940 . . . . . . . . . . . . . . 15 (𝑘 ∈ V → 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹𝑘 / 𝑛(𝑛 + 1)))
5958elv 3475 . . . . . . . . . . . . . 14 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹𝑘 / 𝑛(𝑛 + 1))
60 csbov1g 7459 . . . . . . . . . . . . . . . 16 (𝑘 ∈ V → 𝑘 / 𝑛(𝑛 + 1) = (𝑘 / 𝑛𝑛 + 1))
6160elv 3475 . . . . . . . . . . . . . . 15 𝑘 / 𝑛(𝑛 + 1) = (𝑘 / 𝑛𝑛 + 1)
6261fveq2i 6894 . . . . . . . . . . . . . 14 (𝐹𝑘 / 𝑛(𝑛 + 1)) = (𝐹‘(𝑘 / 𝑛𝑛 + 1))
63 vex 3473 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
6463csbvargi 4428 . . . . . . . . . . . . . . . 16 𝑘 / 𝑛𝑛 = 𝑘
6564oveq1i 7424 . . . . . . . . . . . . . . 15 (𝑘 / 𝑛𝑛 + 1) = (𝑘 + 1)
6665fveq2i 6894 . . . . . . . . . . . . . 14 (𝐹‘(𝑘 / 𝑛𝑛 + 1)) = (𝐹‘(𝑘 + 1))
6759, 62, 663eqtri 2759 . . . . . . . . . . . . 13 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))
6857, 67sseq12i 4008 . . . . . . . . . . . 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 350 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1))) ↔ ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))))
7148, 70bitr4i 278 . . . . . . . . 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 4176 . . . . . . . 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 583 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7642, 44, 753eqtrd 2771 . . . . 5 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))
7776exp31 419 . . . 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 12246 . 2 (𝑖 ∈ ℕ → (𝜑 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖)))
8079impcom 407 1 ((𝜑𝑖 ∈ ℕ) → 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  [wsb 2060  wcel 2099  Vcvv 3469  [wsbc 3774  csb 3889  cun 3942  wss 3944  {csn 4624   ciun 4991   Fn wfn 6537  cfv 6542  (class class class)co 7414  1c1 11125   + caddc 11127  cn 12228  cz 12574  cuz 12838  ...cfz 13502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-n0 12489  df-z 12575  df-uz 12839  df-fz 13503
This theorem is referenced by:  meascnbl  33761
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