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Theorem wlimeq12 33380
 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 = 𝑆𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))

Proof of Theorem wlimeq12
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 488 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵) → 𝐴 = 𝐵)
2 simpl 486 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵) → 𝑅 = 𝑆)
31, 1, 2infeq123d 8991 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵) → inf(𝐴, 𝐴, 𝑅) = inf(𝐵, 𝐵, 𝑆))
43neeq2d 3011 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ↔ 𝑥 ≠ inf(𝐵, 𝐵, 𝑆)))
5 equid 2019 . . . . . . 7 𝑥 = 𝑥
6 predeq123 6132 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝑥 = 𝑥) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
75, 6mp3an3 1447 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
87, 1, 2supeq123d 8960 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵) → sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))
98eqeq2d 2769 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) ↔ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆)))
104, 9anbi12d 633 . . 3 ((𝑅 = 𝑆𝐴 = 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))))
111, 10rabeqbidv 3398 . 2 ((𝑅 = 𝑆𝐴 = 𝐵) → {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))} = {𝑥𝐵 ∣ (𝑥 ≠ inf(𝐵, 𝐵, 𝑆) ∧ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))})
12 df-wlim 33374 . 2 WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
13 df-wlim 33374 . 2 WLim(𝑆, 𝐵) = {𝑥𝐵 ∣ (𝑥 ≠ inf(𝐵, 𝐵, 𝑆) ∧ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))}
1411, 12, 133eqtr4g 2818 1 ((𝑅 = 𝑆𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ≠ wne 2951  {crab 3074  Predcpred 6130  supcsup 8950  infcinf 8951  WLimcwlim 33372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-xp 5534  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-sup 8952  df-inf 8953  df-wlim 33374 This theorem is referenced by:  wlimeq1  33381  wlimeq2  33382
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