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Theorem nfwlim 33183
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 33174 . 2 WLim(𝑅, 𝐴) = {𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
2 nfcv 2979 . . . . 5 𝑥𝑦
3 nfwlim.2 . . . . . 6 𝑥𝐴
4 nfwlim.1 . . . . . 6 𝑥𝑅
53, 3, 4nfinf 8934 . . . . 5 𝑥inf(𝐴, 𝐴, 𝑅)
62, 5nfne 3111 . . . 4 𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅)
74, 3, 2nfpred 6131 . . . . . 6 𝑥Pred(𝑅, 𝐴, 𝑦)
87, 3, 4nfsup 8903 . . . . 5 𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
98nfeq2 2996 . . . 4 𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
106, 9nfan 1900 . . 3 𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))
1110, 3nfrabw 3366 . 2 𝑥{𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
121, 11nfcxfr 2977 1 𝑥WLim(𝑅, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wnfc 2960  wne 3011  {crab 3134  Predcpred 6125  supcsup 8892  infcinf 8893  WLimcwlim 33172
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-sup 8894  df-inf 8895  df-wlim 33174
This theorem is referenced by: (None)
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