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Theorem nfwlim 27896
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 27887 . 2  |- WLim ( R ,  A )  =  { y  e.  A  |  ( y  =/= 
sup ( A ,  A ,  `' R
)  /\  y  =  sup ( Pred ( R ,  A ,  y ) ,  A ,  R ) ) }
2 nfcv 2613 . . . . 5  |-  F/_ x
y
3 nfwlim.2 . . . . . 6  |-  F/_ x A
4 nfwlim.1 . . . . . . 7  |-  F/_ x R
54nfcnv 5119 . . . . . 6  |-  F/_ x `' R
63, 3, 5nfsup 7805 . . . . 5  |-  F/_ x sup ( A ,  A ,  `' R )
72, 6nfne 2779 . . . 4  |-  F/ x  y  =/=  sup ( A ,  A ,  `' R )
84, 3, 2nfpred 27767 . . . . . 6  |-  F/_ x Pred ( R ,  A ,  y )
98, 3, 4nfsup 7805 . . . . 5  |-  F/_ x sup ( Pred ( R ,  A ,  y ) ,  A ,  R )
109nfeq2 2629 . . . 4  |-  F/ x  y  =  sup ( Pred ( R ,  A ,  y ) ,  A ,  R )
117, 10nfan 1863 . . 3  |-  F/ x
( y  =/=  sup ( A ,  A ,  `' R )  /\  y  =  sup ( Pred ( R ,  A , 
y ) ,  A ,  R ) )
1211, 3nfrab 3001 . 2  |-  F/_ x { y  e.  A  |  ( y  =/= 
sup ( A ,  A ,  `' R
)  /\  y  =  sup ( Pred ( R ,  A ,  y ) ,  A ,  R ) ) }
131, 12nfcxfr 2611 1  |-  F/_ xWLim ( R ,  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370   F/_wnfc 2599    =/= wne 2644   {crab 2799   `'ccnv 4940   supcsup 7794   Predcpred 27761  WLimcwlim 27885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-xp 4947  df-cnv 4949  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-sup 7795  df-pred 27762  df-wlim 27887
This theorem is referenced by: (None)
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