Theorem nfwlim 33743

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	⊢ Ⅎ𝑥𝑅
nfwlim.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfwlim	⊢ Ⅎ𝑥WLim(𝑅, 𝐴)

Proof of Theorem nfwlim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wlim 33734	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
2		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfwlim.2	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4		nfwlim.1	. . . . . 6 ⊢ Ⅎ𝑥𝑅
5	3, 3, 4	nfinf 9171	. . . . 5 ⊢ Ⅎ𝑥inf(𝐴, 𝐴, 𝑅)
6	2, 5	nfne 3044	. . . 4 ⊢ Ⅎ𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅)
7	4, 3, 2	nfpred 6196	. . . . . 6 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑦)
8	7, 3, 4	nfsup 9140	. . . . 5 ⊢ Ⅎ𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
9	8	nfeq2 2923	. . . 4 ⊢ Ⅎ𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
10	6, 9	nfan 1903	. . 3 ⊢ Ⅎ𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))
11	10, 3	nfrabw 3311	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
12	1, 11	nfcxfr 2904	1 ⊢ Ⅎ𝑥WLim(𝑅, 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1539  Ⅎwnfc 2886   ≠ wne 2942  {crab 3067  Predcpred 6190  supcsup 9129  infcinf 9130  WLimcwlim 33732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-sup 9131  df-inf 9132  df-wlim 33734

This theorem is referenced by: (None)