Theorem nfwlim 36002

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 35993	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
2		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfwlim.2	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4		nfwlim.1	. . . . . 6 ⊢ Ⅎ𝑥𝑅
5	3, 3, 4	nfinf 9396	. . . . 5 ⊢ Ⅎ𝑥inf(𝐴, 𝐴, 𝑅)
6	2, 5	nfne 3033	. . . 4 ⊢ Ⅎ𝑥 𝑦 ≠ inf(𝐴, 𝐴, 𝑅)
7	4, 3, 2	nfpred 6270	. . . . . 6 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑦)
8	7, 3, 4	nfsup 9364	. . . . 5 ⊢ Ⅎ𝑥sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
9	8	nfeq2 2916	. . . 4 ⊢ Ⅎ𝑥 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅)
10	6, 9	nfan 1901	. . 3 ⊢ Ⅎ𝑥(𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))
11	10, 3	nfrabw 3426	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ (𝑦 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑦 = sup(Pred(𝑅, 𝐴, 𝑦), 𝐴, 𝑅))}
12	1, 11	nfcxfr 2896	1 ⊢ Ⅎ𝑥WLim(𝑅, 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1542  Ⅎwnfc 2883   ≠ wne 2932  {crab 3389  Predcpred 6264  supcsup 9353  infcinf 9354  WLimcwlim 35991

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-sup 9355  df-inf 9356  df-wlim 35993

This theorem is referenced by: (None)