Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  resunimafz0 Structured version   Visualization version   GIF version

Theorem resunimafz0 40368
 Description: Formerly part of proof of eupth2lem3 41404: The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
Hypotheses
Ref Expression
resunimafz0.i (𝜑 → Fun 𝐼)
resunimafz0.f (𝜑𝐹:(0..^(#‘𝐹))⟶dom 𝐼)
resunimafz0.n (𝜑𝑁 ∈ (0..^(#‘𝐹)))
Assertion
Ref Expression
resunimafz0 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))

Proof of Theorem resunimafz0
StepHypRef Expression
1 imaundi 5464 . . . . 5 (𝐹 “ ((0..^𝑁) ∪ {𝑁})) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁}))
2 resunimafz0.n . . . . . . . . 9 (𝜑𝑁 ∈ (0..^(#‘𝐹)))
3 elfzonn0 12380 . . . . . . . . 9 (𝑁 ∈ (0..^(#‘𝐹)) → 𝑁 ∈ ℕ0)
42, 3syl 17 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
5 elnn0uz 11601 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
64, 5sylib 207 . . . . . . 7 (𝜑𝑁 ∈ (ℤ‘0))
7 fzisfzounsn 12445 . . . . . . 7 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
86, 7syl 17 . . . . . 6 (𝜑 → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
98imaeq2d 5385 . . . . 5 (𝜑 → (𝐹 “ (0...𝑁)) = (𝐹 “ ((0..^𝑁) ∪ {𝑁})))
10 resunimafz0.f . . . . . . . 8 (𝜑𝐹:(0..^(#‘𝐹))⟶dom 𝐼)
1110ffnd 5959 . . . . . . 7 (𝜑𝐹 Fn (0..^(#‘𝐹)))
12 fnsnfv 6168 . . . . . . 7 ((𝐹 Fn (0..^(#‘𝐹)) ∧ 𝑁 ∈ (0..^(#‘𝐹))) → {(𝐹𝑁)} = (𝐹 “ {𝑁}))
1311, 2, 12syl2anc 691 . . . . . 6 (𝜑 → {(𝐹𝑁)} = (𝐹 “ {𝑁}))
1413uneq2d 3729 . . . . 5 (𝜑 → ((𝐹 “ (0..^𝑁)) ∪ {(𝐹𝑁)}) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁})))
151, 9, 143eqtr4a 2670 . . . 4 (𝜑 → (𝐹 “ (0...𝑁)) = ((𝐹 “ (0..^𝑁)) ∪ {(𝐹𝑁)}))
1615reseq2d 5317 . . 3 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹𝑁)})))
17 resundi 5330 . . 3 (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹𝑁)}))
1816, 17syl6eq 2660 . 2 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹𝑁)})))
19 resunimafz0.i . . . . 5 (𝜑 → Fun 𝐼)
20 funfn 5833 . . . . 5 (Fun 𝐼𝐼 Fn dom 𝐼)
2119, 20sylib 207 . . . 4 (𝜑𝐼 Fn dom 𝐼)
2210, 2ffvelrnd 6268 . . . 4 (𝜑 → (𝐹𝑁) ∈ dom 𝐼)
23 fnressn 6330 . . . 4 ((𝐼 Fn dom 𝐼 ∧ (𝐹𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹𝑁)}) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
2421, 22, 23syl2anc 691 . . 3 (𝜑 → (𝐼 ↾ {(𝐹𝑁)}) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
2524uneq2d 3729 . 2 (𝜑 → ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
2618, 25eqtrd 2644 1 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  {csn 4125  ⟨cop 4131  dom cdm 5038   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ0cn0 11169  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335 This theorem is referenced by:  trlsegvdeg  41395
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