Theorem nfwlim 35783

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 35774	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
2		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfwlim.2	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4		nfwlim.1	. . . . . 6 ⊢ Ⅎ𝑥𝑅
5	3, 3, 4	nfinf 9410	. . . . 5 ⊢ Ⅎ𝑥inf(𝐴, 𝐴, 𝑅)
6	2, 5	nfne 3026	. . . 4 ⊢ Ⅎ𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅)
7	4, 3, 2	nfpred 6267	. . . . . 6 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑦)
8	7, 3, 4	nfsup 9378	. . . . 5 ⊢ Ⅎ𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
9	8	nfeq2 2909	. . . 4 ⊢ Ⅎ𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
10	6, 9	nfan 1899	. . 3 ⊢ Ⅎ𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))
11	10, 3	nfrabw 3440	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
12	1, 11	nfcxfr 2889	1 ⊢ Ⅎ𝑥WLim(𝑅, 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1540  Ⅎwnfc 2876   ≠ wne 2925  {crab 3402  Predcpred 6261  supcsup 9367  infcinf 9368  WLimcwlim 35772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-sup 9369  df-inf 9370  df-wlim 35774

This theorem is referenced by: (None)