Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wlimeq12 Structured version   Unicode version

Theorem wlimeq12 27755
Description: Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
Assertion
Ref Expression
wlimeq12  |-  ( ( R  =  S  /\  A  =  B )  -> WLim ( R ,  A
)  = WLim ( S ,  B ) )

Proof of Theorem wlimeq12
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . 3  |-  ( ( R  =  S  /\  A  =  B )  ->  A  =  B )
2 cnveq 5012 . . . . . . 7  |-  ( R  =  S  ->  `' R  =  `' S
)
32adantr 465 . . . . . 6  |-  ( ( R  =  S  /\  A  =  B )  ->  `' R  =  `' S )
41, 1, 3supeq123d 7699 . . . . 5  |-  ( ( R  =  S  /\  A  =  B )  ->  sup ( A ,  A ,  `' R
)  =  sup ( B ,  B ,  `' S ) )
54neeq2d 2621 . . . 4  |-  ( ( R  =  S  /\  A  =  B )  ->  ( x  =/=  sup ( A ,  A ,  `' R )  <->  x  =/=  sup ( B ,  B ,  `' S ) ) )
6 eqid 2442 . . . . . . 7  |-  x  =  x
7 predeq123 27625 . . . . . . 7  |-  ( ( R  =  S  /\  A  =  B  /\  x  =  x )  ->  Pred ( R ,  A ,  x )  =  Pred ( S ,  B ,  x )
)
86, 7mp3an3 1303 . . . . . 6  |-  ( ( R  =  S  /\  A  =  B )  ->  Pred ( R ,  A ,  x )  =  Pred ( S ,  B ,  x )
)
9 simpl 457 . . . . . 6  |-  ( ( R  =  S  /\  A  =  B )  ->  R  =  S )
108, 1, 9supeq123d 7699 . . . . 5  |-  ( ( R  =  S  /\  A  =  B )  ->  sup ( Pred ( R ,  A ,  x ) ,  A ,  R )  =  sup ( Pred ( S ,  B ,  x ) ,  B ,  S ) )
1110eqeq2d 2453 . . . 4  |-  ( ( R  =  S  /\  A  =  B )  ->  ( x  =  sup ( Pred ( R ,  A ,  x ) ,  A ,  R )  <-> 
x  =  sup ( Pred ( S ,  B ,  x ) ,  B ,  S ) ) )
125, 11anbi12d 710 . . 3  |-  ( ( R  =  S  /\  A  =  B )  ->  ( ( x  =/= 
sup ( A ,  A ,  `' R
)  /\  x  =  sup ( Pred ( R ,  A ,  x
) ,  A ,  R ) )  <->  ( x  =/=  sup ( B ,  B ,  `' S
)  /\  x  =  sup ( Pred ( S ,  B ,  x
) ,  B ,  S ) ) ) )
131, 12rabeqbidv 2966 . 2  |-  ( ( R  =  S  /\  A  =  B )  ->  { x  e.  A  |  ( x  =/= 
sup ( A ,  A ,  `' R
)  /\  x  =  sup ( Pred ( R ,  A ,  x
) ,  A ,  R ) ) }  =  { x  e.  B  |  ( x  =/=  sup ( B ,  B ,  `' S )  /\  x  =  sup ( Pred ( S ,  B ,  x ) ,  B ,  S ) ) } )
14 df-wlim 27749 . 2  |- WLim ( R ,  A )  =  { x  e.  A  |  ( x  =/= 
sup ( A ,  A ,  `' R
)  /\  x  =  sup ( Pred ( R ,  A ,  x
) ,  A ,  R ) ) }
15 df-wlim 27749 . 2  |- WLim ( S ,  B )  =  { x  e.  B  |  ( x  =/= 
sup ( B ,  B ,  `' S
)  /\  x  =  sup ( Pred ( S ,  B ,  x
) ,  B ,  S ) ) }
1613, 14, 153eqtr4g 2499 1  |-  ( ( R  =  S  /\  A  =  B )  -> WLim ( R ,  A
)  = WLim ( S ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    =/= wne 2605   {crab 2718   `'ccnv 4838   supcsup 7689   Predcpred 27623  WLimcwlim 27747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-sup 7690  df-pred 27624  df-wlim 27749
This theorem is referenced by:  wlimeq1  27756  wlimeq2  27757
  Copyright terms: Public domain W3C validator