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Theorem iuninc 29702
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 6880 . . . . . 6 (𝑗 = 1 → (1...𝑗) = (1...1))
21iuneq1d 4735 . . . . 5 (𝑗 = 1 → 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...1)(𝐹𝑛))
3 fveq2 6406 . . . . 5 (𝑗 = 1 → (𝐹𝑗) = (𝐹‘1))
42, 3eqeq12d 2819 . . . 4 (𝑗 = 1 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)))
54imbi2d 331 . . 3 (𝑗 = 1 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1))))
6 oveq2 6880 . . . . . 6 (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
76iuneq1d 4735 . . . . 5 (𝑗 = 𝑘 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
8 fveq2 6406 . . . . 5 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
97, 8eqeq12d 2819 . . . 4 (𝑗 = 𝑘 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)))
109imbi2d 331 . . 3 (𝑗 = 𝑘 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘))))
11 oveq2 6880 . . . . . 6 (𝑗 = (𝑘 + 1) → (1...𝑗) = (1...(𝑘 + 1)))
1211iuneq1d 4735 . . . . 5 (𝑗 = (𝑘 + 1) → 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
13 fveq2 6406 . . . . 5 (𝑗 = (𝑘 + 1) → (𝐹𝑗) = (𝐹‘(𝑘 + 1)))
1412, 13eqeq12d 2819 . . . 4 (𝑗 = (𝑘 + 1) → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1))))
1514imbi2d 331 . . 3 (𝑗 = (𝑘 + 1) → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
16 oveq2 6880 . . . . . 6 (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
1716iuneq1d 4735 . . . . 5 (𝑗 = 𝑖 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...𝑖)(𝐹𝑛))
18 fveq2 6406 . . . . 5 (𝑗 = 𝑖 → (𝐹𝑗) = (𝐹𝑖))
1917, 18eqeq12d 2819 . . . 4 (𝑗 = 𝑖 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖)))
2019imbi2d 331 . . 3 (𝑗 = 𝑖 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))))
21 1z 11671 . . . . . . 7 1 ∈ ℤ
22 fzsn 12604 . . . . . . 7 (1 ∈ ℤ → (1...1) = {1})
2321, 22ax-mp 5 . . . . . 6 (1...1) = {1}
24 iuneq1 4724 . . . . . 6 ((1...1) = {1} → 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛))
2523, 24ax-mp 5 . . . . 5 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛)
26 1ex 10319 . . . . . 6 1 ∈ V
27 fveq2 6406 . . . . . 6 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
2826, 27iunxsn 4793 . . . . 5 𝑛 ∈ {1} (𝐹𝑛) = (𝐹‘1)
2925, 28eqtri 2826 . . . 4 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)
3029a1i 11 . . 3 (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1))
31 simpll 774 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑘 ∈ ℕ)
32 elnnuz 11940 . . . . . . . . . 10 (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ‘1))
33 fzsuc 12609 . . . . . . . . . 10 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
3432, 33sylbi 208 . . . . . . . . 9 (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
3534iuneq1d 4735 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
36 iunxun 4795 . . . . . . . . 9 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛))
37 ovex 6904 . . . . . . . . . . 11 (𝑘 + 1) ∈ V
38 fveq2 6406 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
3937, 38iunxsn 4793 . . . . . . . . . 10 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛) = (𝐹‘(𝑘 + 1))
4039uneq2i 3961 . . . . . . . . 9 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛)) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
4136, 40eqtri 2826 . . . . . . . 8 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
4235, 41syl6eq 2854 . . . . . . 7 (𝑘 ∈ ℕ → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))))
4331, 42syl 17 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))))
44 simpr 473 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘))
4544uneq1d 3963 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))))
46 simplr 776 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝜑)
47 iuninc.2 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
4847sbt 2578 . . . . . . . . 9 [𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
49 sbim 2554 . . . . . . . . . 10 ([𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))))
50 sban 2558 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ))
51 nfv 2005 . . . . . . . . . . . . . 14 𝑛𝜑
5251sbf 2539 . . . . . . . . . . . . 13 ([𝑘 / 𝑛]𝜑𝜑)
53 clelsb3 2911 . . . . . . . . . . . . 13 ([𝑘 / 𝑛]𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)
5452, 53anbi12i 614 . . . . . . . . . . . 12 (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ) ↔ (𝜑𝑘 ∈ ℕ))
5550, 54bitr2i 267 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) ↔ [𝑘 / 𝑛](𝜑𝑛 ∈ ℕ))
56 sbsbc 3635 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
57 vex 3392 . . . . . . . . . . . . 13 𝑘 ∈ V
58 sbcssg 4276 . . . . . . . . . . . . 13 (𝑘 ∈ V → ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ 𝑘 / 𝑛(𝐹𝑛) ⊆ 𝑘 / 𝑛(𝐹‘(𝑛 + 1))))
5957, 58ax-mp 5 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ 𝑘 / 𝑛(𝐹𝑛) ⊆ 𝑘 / 𝑛(𝐹‘(𝑛 + 1)))
60 csbfv 6451 . . . . . . . . . . . . 13 𝑘 / 𝑛(𝐹𝑛) = (𝐹𝑘)
61 csbfv2g 6450 . . . . . . . . . . . . . . 15 (𝑘 ∈ V → 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹𝑘 / 𝑛(𝑛 + 1)))
6257, 61ax-mp 5 . . . . . . . . . . . . . 14 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹𝑘 / 𝑛(𝑛 + 1))
63 csbov1g 6916 . . . . . . . . . . . . . . . 16 (𝑘 ∈ V → 𝑘 / 𝑛(𝑛 + 1) = (𝑘 / 𝑛𝑛 + 1))
6457, 63ax-mp 5 . . . . . . . . . . . . . . 15 𝑘 / 𝑛(𝑛 + 1) = (𝑘 / 𝑛𝑛 + 1)
6564fveq2i 6409 . . . . . . . . . . . . . 14 (𝐹𝑘 / 𝑛(𝑛 + 1)) = (𝐹‘(𝑘 / 𝑛𝑛 + 1))
6657csbvargi 4199 . . . . . . . . . . . . . . . 16 𝑘 / 𝑛𝑛 = 𝑘
6766oveq1i 6882 . . . . . . . . . . . . . . 15 (𝑘 / 𝑛𝑛 + 1) = (𝑘 + 1)
6867fveq2i 6409 . . . . . . . . . . . . . 14 (𝐹‘(𝑘 / 𝑛𝑛 + 1)) = (𝐹‘(𝑘 + 1))
6962, 65, 683eqtri 2830 . . . . . . . . . . . . 13 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))
7060, 69sseq12i 3826 . . . . . . . . . . . 12 (𝑘 / 𝑛(𝐹𝑛) ⊆ 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) ↔ (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)))
7156, 59, 703bitrri 289 . . . . . . . . . . 11 ((𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
7255, 71imbi12i 341 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1))) ↔ ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))))
7349, 72bitr4i 269 . . . . . . . . 9 ([𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1))))
7448, 73mpbi 221 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)))
75 ssequn1 3980 . . . . . . . 8 ((𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7674, 75sylib 209 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7746, 31, 76syl2anc 575 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7843, 45, 773eqtrd 2842 . . . . 5 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))
7978exp31 408 . . . 4 (𝑘 ∈ ℕ → (𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
8079a2d 29 . . 3 (𝑘 ∈ ℕ → ((𝜑 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → (𝜑 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
815, 10, 15, 20, 30, 80nnind 11321 . 2 (𝑖 ∈ ℕ → (𝜑 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖)))
8281impcom 396 1 ((𝜑𝑖 ∈ ℕ) → 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  [wsb 2060  wcel 2156  Vcvv 3389  [wsbc 3631  csb 3726  cun 3765  wss 3767  {csn 4368   ciun 4710   Fn wfn 6094  cfv 6099  (class class class)co 6872  1c1 10220   + caddc 10222  cn 11303  cz 11641  cuz 11902  ...cfz 12547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177  ax-cnex 10275  ax-resscn 10276  ax-1cn 10277  ax-icn 10278  ax-addcl 10279  ax-addrcl 10280  ax-mulcl 10281  ax-mulrcl 10282  ax-mulcom 10283  ax-addass 10284  ax-mulass 10285  ax-distr 10286  ax-i2m1 10287  ax-1ne0 10288  ax-1rid 10289  ax-rnegex 10290  ax-rrecex 10291  ax-cnre 10292  ax-pre-lttri 10293  ax-pre-lttrn 10294  ax-pre-ltadd 10295  ax-pre-mulgt0 10296
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-nel 3080  df-ral 3099  df-rex 3100  df-reu 3101  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5217  df-eprel 5222  df-po 5230  df-so 5231  df-fr 5268  df-we 5270  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-pred 5891  df-ord 5937  df-on 5938  df-lim 5939  df-suc 5940  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6833  df-ov 6875  df-oprab 6876  df-mpt2 6877  df-om 7294  df-1st 7396  df-2nd 7397  df-wrecs 7640  df-recs 7702  df-rdg 7740  df-er 7977  df-en 8191  df-dom 8192  df-sdom 8193  df-pnf 10359  df-mnf 10360  df-xr 10361  df-ltxr 10362  df-le 10363  df-sub 10551  df-neg 10552  df-nn 11304  df-n0 11558  df-z 11642  df-uz 11903  df-fz 12548
This theorem is referenced by:  meascnbl  30605
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