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Theorem nfwlim 34267
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 34258 . 2 WLim(𝑅, 𝐴) = {𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
2 nfcv 2905 . . . . 5 𝑥𝑦
3 nfwlim.2 . . . . . 6 𝑥𝐴
4 nfwlim.1 . . . . . 6 𝑥𝑅
53, 3, 4nfinf 9414 . . . . 5 𝑥inf(𝐴, 𝐴, 𝑅)
62, 5nfne 3043 . . . 4 𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅)
74, 3, 2nfpred 6256 . . . . . 6 𝑥Pred(𝑅, 𝐴, 𝑦)
87, 3, 4nfsup 9383 . . . . 5 𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
98nfeq2 2922 . . . 4 𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
106, 9nfan 1902 . . 3 𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))
1110, 3nfrabw 3438 . 2 𝑥{𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
121, 11nfcxfr 2903 1 𝑥WLim(𝑅, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wnfc 2885  wne 2941  {crab 3405  Predcpred 6250  supcsup 9372  infcinf 9373  WLimcwlim 34256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-sup 9374  df-inf 9375  df-wlim 34258
This theorem is referenced by: (None)
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