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Theorem wlimeq12 29549
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 5186 . . . . . . 7  |-  ( R  =  S  ->  `' R  =  `' S
)
32adantr 465 . . . . . 6  |-  ( ( R  =  S  /\  A  =  B )  ->  `' R  =  `' S )
41, 1, 3supeq123d 7927 . . . . 5  |-  ( ( R  =  S  /\  A  =  B )  ->  sup ( A ,  A ,  `' R
)  =  sup ( B ,  B ,  `' S ) )
54neeq2d 2735 . . . 4  |-  ( ( R  =  S  /\  A  =  B )  ->  ( x  =/=  sup ( A ,  A ,  `' R )  <->  x  =/=  sup ( B ,  B ,  `' S ) ) )
6 equid 1792 . . . . . . 7  |-  x  =  x
7 predeq123 29419 . . . . . . 7  |-  ( ( R  =  S  /\  A  =  B  /\  x  =  x )  ->  Pred ( R ,  A ,  x )  =  Pred ( S ,  B ,  x )
)
86, 7mp3an3 1313 . . . . . 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 7927 . . . . 5  |-  ( ( R  =  S  /\  A  =  B )  ->  sup ( Pred ( R ,  A ,  x ) ,  A ,  R )  =  sup ( Pred ( S ,  B ,  x ) ,  B ,  S ) )
1110eqeq2d 2471 . . . 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 3104 . 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 29543 . 2  |- WLim ( R ,  A )  =  { x  e.  A  |  ( x  =/= 
sup ( A ,  A ,  `' R
)  /\  x  =  sup ( Pred ( R ,  A ,  x
) ,  A ,  R ) ) }
15 df-wlim 29543 . 2  |- WLim ( S ,  B )  =  { x  e.  B  |  ( x  =/= 
sup ( B ,  B ,  `' S
)  /\  x  =  sup ( Pred ( S ,  B ,  x
) ,  B ,  S ) ) }
1613, 14, 153eqtr4g 2523 1  |-  ( ( R  =  S  /\  A  =  B )  -> WLim ( R ,  A
)  = WLim ( S ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    =/= wne 2652   {crab 2811   `'ccnv 5007   supcsup 7918   Predcpred 29417  WLimcwlim 29541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-sup 7919  df-pred 29418  df-wlim 29543
This theorem is referenced by:  wlimeq1  29550  wlimeq2  29551
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