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Theorem nfwlim 32095
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 32086 . 2 WLim(𝑅, 𝐴) = {𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
2 nfcv 2903 . . . . 5 𝑥𝑦
3 nfwlim.2 . . . . . 6 𝑥𝐴
4 nfwlim.1 . . . . . 6 𝑥𝑅
53, 3, 4nfinf 8556 . . . . 5 𝑥inf(𝐴, 𝐴, 𝑅)
62, 5nfne 3033 . . . 4 𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅)
74, 3, 2nfpred 5847 . . . . . 6 𝑥Pred(𝑅, 𝐴, 𝑦)
87, 3, 4nfsup 8525 . . . . 5 𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
98nfeq2 2919 . . . 4 𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
106, 9nfan 1978 . . 3 𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))
1110, 3nfrab 3263 . 2 𝑥{𝑦𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
121, 11nfcxfr 2901 1 𝑥WLim(𝑅, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wnfc 2890  wne 2933  {crab 3055  Predcpred 5841  supcsup 8514  infcinf 8515  WLimcwlim 32084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-xp 5273  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-sup 8516  df-inf 8517  df-wlim 32086
This theorem is referenced by: (None)
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