Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nfwlim Structured version   Visualization version   GIF version

Theorem nfwlim 31012
 Description: Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
Hypotheses
Ref Expression
nfwlim.1 𝑥𝑅
nfwlim.2 𝑥𝐴
Assertion
Ref Expression
nfwlim 𝑥WLim(𝑅, 𝐴)

Proof of Theorem nfwlim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-wlim 31002 . 2 WLim(𝑅, 𝐴) = {𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
2 nfcv 2751 . . . . 5 𝑥𝑦
3 nfwlim.2 . . . . . 6 𝑥𝐴
4 nfwlim.1 . . . . . 6 𝑥𝑅
53, 3, 4nfinf 8271 . . . . 5 𝑥inf(𝐴, 𝐴, 𝑅)
62, 5nfne 2882 . . . 4 𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅)
74, 3, 2nfpred 5602 . . . . . 6 𝑥Pred(𝑅, 𝐴, 𝑦)
87, 3, 4nfsup 8240 . . . . 5 𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
98nfeq2 2766 . . . 4 𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
106, 9nfan 1816 . . 3 𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))
1110, 3nfrab 3100 . 2 𝑥{𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
121, 11nfcxfr 2749 1 𝑥WLim(𝑅, 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  Ⅎwnfc 2738   ≠ wne 2780  {crab 2900  Predcpred 5596  supcsup 8229  infcinf 8230  WLimcwlim 30998 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-sup 8231  df-inf 8232  df-wlim 31002 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator