Theorem nfwlim 36584

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 36575	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
2		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfwlim.2	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4		nfwlim.1	. . . . . 6 ⊢ Ⅎ𝑥𝑅
5	3, 3, 4	nfinf 9475	. . . . 5 ⊢ Ⅎ𝑥inf(𝐴, 𝐴, 𝑅)
6	2, 5	nfne 3058	. . . 4 ⊢ Ⅎ𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅)
7	4, 3, 2	nfpred 6309	. . . . . 6 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑦)
8	7, 3, 4	nfsup 9443	. . . . 5 ⊢ Ⅎ𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
9	8	nfeq2 2939	. . . 4 ⊢ Ⅎ𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
10	6, 9	nfan 1931	. . 3 ⊢ Ⅎ𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))
11	10, 3	nfrabw 3447	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
12	1, 11	nfcxfr 2920	1 ⊢ Ⅎ𝑥WLim(𝑅, 𝐴)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  Ⅎwnfc 2907   ≠ wne 2955  {crab 3412  Predcpred 6303  supcsup 9432  infcinf 9433  WLimcwlim 36573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-sup 9434  df-inf 9435  df-wlim 36575

This theorem is used by: (None)