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Theorem nfwlim 29305
Description: Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
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.
StepHypRef Expression
1 df-wlim 29296 . 2  |- WLim ( R ,  A )  =  { y  e.  A  |  ( y  =/= 
sup ( A ,  A ,  `' R
)  /\  y  =  sup ( Pred ( R ,  A ,  y ) ,  A ,  R ) ) }
2 nfcv 2629 . . . . 5  |-  F/_ x
y
3 nfwlim.2 . . . . . 6  |-  F/_ x A
4 nfwlim.1 . . . . . . 7  |-  F/_ x R
54nfcnv 5187 . . . . . 6  |-  F/_ x `' R
63, 3, 5nfsup 7923 . . . . 5  |-  F/_ x sup ( A ,  A ,  `' R )
72, 6nfne 2798 . . . 4  |-  F/ x  y  =/=  sup ( A ,  A ,  `' R )
84, 3, 2nfpred 29176 . . . . . 6  |-  F/_ x Pred ( R ,  A ,  y )
98, 3, 4nfsup 7923 . . . . 5  |-  F/_ x sup ( Pred ( R ,  A ,  y ) ,  A ,  R )
109nfeq2 2646 . . . 4  |-  F/ x  y  =  sup ( Pred ( R ,  A ,  y ) ,  A ,  R )
117, 10nfan 1875 . . 3  |-  F/ x
( y  =/=  sup ( A ,  A ,  `' R )  /\  y  =  sup ( Pred ( R ,  A , 
y ) ,  A ,  R ) )
1211, 3nfrab 3048 . 2  |-  F/_ x { y  e.  A  |  ( y  =/= 
sup ( A ,  A ,  `' R
)  /\  y  =  sup ( Pred ( R ,  A ,  y ) ,  A ,  R ) ) }
131, 12nfcxfr 2627 1  |-  F/_ xWLim ( R ,  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   F/_wnfc 2615    =/= wne 2662   {crab 2821   `'ccnv 5004   supcsup 7912   Predcpred 29170  WLimcwlim 29294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-sup 7913  df-pred 29171  df-wlim 29296
This theorem is referenced by: (None)
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