Theorem nfwlim 31768

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	|- F/_ x R
nfwlim.2	|- F/_ x A

Assertion

Ref	Expression
nfwlim	|- F/_ xWLim ( R , A )

Proof of Theorem nfwlim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wlim 31758	. 2 |- WLim ( R , A ) = { y e. A | ( y =/= inf ( A , A , R ) /\ y = sup ( Pred ( R , A , y ) , A , R ) ) }
2		nfcv 2764	. . . . 5 |- F/_ x y
3		nfwlim.2	. . . . . 6 |- F/_ x A
4		nfwlim.1	. . . . . 6 |- F/_ x R
5	3, 3, 4	nfinf 8388	. . . . 5 |- F/_ xinf ( A , A , R )
6	2, 5	nfne 2894	. . . 4 |- F/ x y =/= inf ( A , A , R )
7	4, 3, 2	nfpred 5685	. . . . . 6 |- F/_ x Pred ( R , A , y )
8	7, 3, 4	nfsup 8357	. . . . 5 |- F/_ x sup ( Pred ( R , A , y ) , A , R )
9	8	nfeq2 2780	. . . 4 |- F/ x y = sup ( Pred ( R , A , y ) , A , R )
10	6, 9	nfan 1828	. . 3 |- F/ x ( y =/= inf ( A , A , R ) /\ y = sup ( Pred ( R , A , y ) , A , R ) )
11	10, 3	nfrab 3123	. 2 |- F/_ x { y e. A | ( y =/= inf ( A , A , R ) /\ y = sup ( Pred ( R , A , y ) , A , R ) ) }
12	1, 11	nfcxfr 2762	1 |- F/_ xWLim ( R , A )

Syntax hints:   /\ wa 384   = wceq 1483   F/_wnfc 2751   =/= wne 2794   {crab 2916   Predcpred 5679   supcsup 8346  infcinf 8347  WLimcwlim 31754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-sup 8348  df-inf 8349  df-wlim 31758

This theorem is referenced by: (None)