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Theorem iuninc 29221
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 6612 . . . . . 6 (𝑗 = 1 → (1...𝑗) = (1...1))
21iuneq1d 4511 . . . . 5 (𝑗 = 1 → 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...1)(𝐹𝑛))
3 fveq2 6148 . . . . 5 (𝑗 = 1 → (𝐹𝑗) = (𝐹‘1))
42, 3eqeq12d 2636 . . . 4 (𝑗 = 1 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)))
54imbi2d 330 . . 3 (𝑗 = 1 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1))))
6 oveq2 6612 . . . . . 6 (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
76iuneq1d 4511 . . . . 5 (𝑗 = 𝑘 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
8 fveq2 6148 . . . . 5 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
97, 8eqeq12d 2636 . . . 4 (𝑗 = 𝑘 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)))
109imbi2d 330 . . 3 (𝑗 = 𝑘 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘))))
11 oveq2 6612 . . . . . 6 (𝑗 = (𝑘 + 1) → (1...𝑗) = (1...(𝑘 + 1)))
1211iuneq1d 4511 . . . . 5 (𝑗 = (𝑘 + 1) → 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
13 fveq2 6148 . . . . 5 (𝑗 = (𝑘 + 1) → (𝐹𝑗) = (𝐹‘(𝑘 + 1)))
1412, 13eqeq12d 2636 . . . 4 (𝑗 = (𝑘 + 1) → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1))))
1514imbi2d 330 . . 3 (𝑗 = (𝑘 + 1) → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
16 oveq2 6612 . . . . . 6 (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
1716iuneq1d 4511 . . . . 5 (𝑗 = 𝑖 𝑛 ∈ (1...𝑗)(𝐹𝑛) = 𝑛 ∈ (1...𝑖)(𝐹𝑛))
18 fveq2 6148 . . . . 5 (𝑗 = 𝑖 → (𝐹𝑗) = (𝐹𝑖))
1917, 18eqeq12d 2636 . . . 4 (𝑗 = 𝑖 → ( 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗) ↔ 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖)))
2019imbi2d 330 . . 3 (𝑗 = 𝑖 → ((𝜑 𝑛 ∈ (1...𝑗)(𝐹𝑛) = (𝐹𝑗)) ↔ (𝜑 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))))
21 1z 11351 . . . . . . 7 1 ∈ ℤ
22 fzsn 12325 . . . . . . 7 (1 ∈ ℤ → (1...1) = {1})
2321, 22ax-mp 5 . . . . . 6 (1...1) = {1}
24 iuneq1 4500 . . . . . 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 9979 . . . . . 6 1 ∈ V
27 fveq2 6148 . . . . . 6 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
2826, 27iunxsn 4569 . . . . 5 𝑛 ∈ {1} (𝐹𝑛) = (𝐹‘1)
2925, 28eqtri 2643 . . . 4 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)
3029a1i 11 . . 3 (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1))
31 simpll 789 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑘 ∈ ℕ)
32 elnnuz 11668 . . . . . . . . . 10 (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ‘1))
33 fzsuc 12330 . . . . . . . . . 10 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
3432, 33sylbi 207 . . . . . . . . 9 (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
3534iuneq1d 4511 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
36 iunxun 4571 . . . . . . . . 9 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛))
37 ovex 6632 . . . . . . . . . . 11 (𝑘 + 1) ∈ V
38 fveq2 6148 . . . . . . . . . . 11 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
3937, 38iunxsn 4569 . . . . . . . . . 10 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛) = (𝐹‘(𝑘 + 1))
4039uneq2i 3742 . . . . . . . . 9 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛)) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
4136, 40eqtri 2643 . . . . . . . 8 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
4235, 41syl6eq 2671 . . . . . . 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 477 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘))
4544uneq1d 3744 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))))
46 simplr 791 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝜑)
47 iuninc.2 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
4847sbt 2418 . . . . . . . . 9 [𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
49 sbim 2394 . . . . . . . . . 10 ([𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))))
50 sban 2398 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ))
51 nfv 1840 . . . . . . . . . . . . . 14 𝑛𝜑
5251sbf 2379 . . . . . . . . . . . . 13 ([𝑘 / 𝑛]𝜑𝜑)
53 clelsb3 2726 . . . . . . . . . . . . 13 ([𝑘 / 𝑛]𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)
5452, 53anbi12i 732 . . . . . . . . . . . 12 (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ ℕ) ↔ (𝜑𝑘 ∈ ℕ))
5550, 54bitr2i 265 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) ↔ [𝑘 / 𝑛](𝜑𝑛 ∈ ℕ))
56 sbsbc 3421 . . . . . . . . . . . 12 ([𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
57 vex 3189 . . . . . . . . . . . . 13 𝑘 ∈ V
58 sbcssg 4057 . . . . . . . . . . . . 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 6190 . . . . . . . . . . . . 13 𝑘 / 𝑛(𝐹𝑛) = (𝐹𝑘)
61 csbfv2g 6189 . . . . . . . . . . . . . . 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 6643 . . . . . . . . . . . . . . . 16 (𝑘 ∈ V → 𝑘 / 𝑛(𝑛 + 1) = (𝑘 / 𝑛𝑛 + 1))
6457, 63ax-mp 5 . . . . . . . . . . . . . . 15 𝑘 / 𝑛(𝑛 + 1) = (𝑘 / 𝑛𝑛 + 1)
6564fveq2i 6151 . . . . . . . . . . . . . 14 (𝐹𝑘 / 𝑛(𝑛 + 1)) = (𝐹‘(𝑘 / 𝑛𝑛 + 1))
66 csbvarg 3975 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ V → 𝑘 / 𝑛𝑛 = 𝑘)
6757, 66ax-mp 5 . . . . . . . . . . . . . . . 16 𝑘 / 𝑛𝑛 = 𝑘
6867oveq1i 6614 . . . . . . . . . . . . . . 15 (𝑘 / 𝑛𝑛 + 1) = (𝑘 + 1)
6968fveq2i 6151 . . . . . . . . . . . . . 14 (𝐹‘(𝑘 / 𝑛𝑛 + 1)) = (𝐹‘(𝑘 + 1))
7062, 65, 693eqtri 2647 . . . . . . . . . . . . 13 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))
7160, 70sseq12i 3610 . . . . . . . . . . . 12 (𝑘 / 𝑛(𝐹𝑛) ⊆ 𝑘 / 𝑛(𝐹‘(𝑛 + 1)) ↔ (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)))
7256, 59, 713bitrri 287 . . . . . . . . . . 11 ((𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
7355, 72imbi12i 340 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1))) ↔ ([𝑘 / 𝑛](𝜑𝑛 ∈ ℕ) → [𝑘 / 𝑛](𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))))
7449, 73bitr4i 267 . . . . . . . . 9 ([𝑘 / 𝑛]((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ↔ ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1))))
7548, 74mpbi 220 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)))
76 ssequn1 3761 . . . . . . . 8 ((𝐹𝑘) ⊆ (𝐹‘(𝑘 + 1)) ↔ ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7775, 76sylib 208 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7846, 31, 77syl2anc 692 . . . . . 6 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → ((𝐹𝑘) ∪ (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
7943, 45, 783eqtrd 2659 . . . . 5 (((𝑘 ∈ ℕ ∧ 𝜑) ∧ 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))
8079exp31 629 . . . 4 (𝑘 ∈ ℕ → (𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
8180a2d 29 . . 3 (𝑘 ∈ ℕ → ((𝜑 𝑛 ∈ (1...𝑘)(𝐹𝑛) = (𝐹𝑘)) → (𝜑 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = (𝐹‘(𝑘 + 1)))))
825, 10, 15, 20, 30, 81nnind 10982 . 2 (𝑖 ∈ ℕ → (𝜑 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖)))
8382impcom 446 1 ((𝜑𝑖 ∈ ℕ) → 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  [wsb 1877  wcel 1987  Vcvv 3186  [wsbc 3417  csb 3514  cun 3553  wss 3555  {csn 4148   ciun 4485   Fn wfn 5842  cfv 5847  (class class class)co 6604  1c1 9881   + caddc 9883  cn 10964  cz 11321  cuz 11631  ...cfz 12268
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  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-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269
This theorem is referenced by:  meascnbl  30060
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