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Theorem nfwlim 31892
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 31883 . 2 WLim(𝑅, 𝐴) = {𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
2 nfcv 2793 . . . . 5 𝑥𝑦
3 nfwlim.2 . . . . . 6 𝑥𝐴
4 nfwlim.1 . . . . . 6 𝑥𝑅
53, 3, 4nfinf 8429 . . . . 5 𝑥inf(𝐴, 𝐴, 𝑅)
62, 5nfne 2923 . . . 4 𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅)
74, 3, 2nfpred 5723 . . . . . 6 𝑥Pred(𝑅, 𝐴, 𝑦)
87, 3, 4nfsup 8398 . . . . 5 𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
98nfeq2 2809 . . . 4 𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
106, 9nfan 1868 . . 3 𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))
1110, 3nfrab 3153 . 2 𝑥{𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
121, 11nfcxfr 2791 1 𝑥WLim(𝑅, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wnfc 2780  wne 2823  {crab 2945  Predcpred 5717  supcsup 8387  infcinf 8388  WLimcwlim 31881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-sup 8389  df-inf 8390  df-wlim 31883
This theorem is referenced by: (None)
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