Theorem wlimeq12 31765

Description: Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)

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.

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( R = S /\ A = B ) -> A = B )
2		simpl 473	. . . . . 6 |- ( ( R = S /\ A = B ) -> R = S )
3	1, 1, 2	infeq123d 8387	. . . . 5 |- ( ( R = S /\ A = B ) -> inf ( A , A , R ) = inf ( B , B , S ) )
4	3	neeq2d 2854	. . . 4 |- ( ( R = S /\ A = B ) -> ( x =/= inf ( A , A , R ) <-> x =/= inf ( B , B , S ) ) )
5		equid 1939	. . . . . . 7 |- x = x
6		predeq123 5681	. . . . . . 7 |- ( ( R = S /\ A = B /\ x = x ) -> Pred ( R , A , x ) = Pred ( S , B , x ) )
7	5, 6	mp3an3 1413	. . . . . 6 |- ( ( R = S /\ A = B ) -> Pred ( R , A , x ) = Pred ( S , B , x ) )
8	7, 1, 2	supeq123d 8356	. . . . 5 |- ( ( R = S /\ A = B ) -> sup ( Pred ( R , A , x ) , A , R ) = sup ( Pred ( S , B , x ) , B , S ) )
9	8	eqeq2d 2632	. . . 4 |- ( ( R = S /\ A = B ) -> ( x = sup ( Pred ( R , A , x ) , A , R ) <-> x = sup ( Pred ( S , B , x ) , B , S ) ) )
10	4, 9	anbi12d 747	. . 3 |- ( ( R = S /\ A = B ) -> ( ( x =/= inf ( A , A , R ) /\ x = sup ( Pred ( R , A , x ) , A , R ) ) <-> ( x =/= inf ( B , B , S ) /\ x = sup ( Pred ( S , B , x ) , B , S ) ) ) )
11	1, 10	rabeqbidv 3195	. 2 |- ( ( R = S /\ A = B ) -> { x e. A | ( x =/= inf ( A , A , R ) /\ x = sup ( Pred ( R , A , x ) , A , R ) ) } = { x e. B | ( x =/= inf ( B , B , S ) /\ x = sup ( Pred ( S , B , x ) , B , S ) ) } )
12		df-wlim 31758	. 2 |- WLim ( R , A ) = { x e. A | ( x =/= inf ( A , A , R ) /\ x = sup ( Pred ( R , A , x ) , A , R ) ) }
13		df-wlim 31758	. 2 |- WLim ( S , B ) = { x e. B | ( x =/= inf ( B , B , S ) /\ x = sup ( Pred ( S , B , x ) , B , S ) ) }
14	11, 12, 13	3eqtr4g 2681	1 |- ( ( R = S /\ A = B ) -> WLim ( R , A ) = WLim ( S , B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= 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

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-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:  wlimeq1  31766  wlimeq2  31767