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Theorem wlimeq12 34433
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 486 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵) → 𝐴 = 𝐵)
2 simpl 484 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵) → 𝑅 = 𝑆)
31, 1, 2infeq123d 9424 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵) → inf(𝐴, 𝐴, 𝑅) = inf(𝐵, 𝐵, 𝑆))
43neeq2d 3005 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ↔ 𝑥 ≠ inf(𝐵, 𝐵, 𝑆)))
5 equid 2016 . . . . . . 7 𝑥 = 𝑥
6 predeq123 6259 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝑥 = 𝑥) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
75, 6mp3an3 1451 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
87, 1, 2supeq123d 9393 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵) → sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))
98eqeq2d 2748 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) ↔ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆)))
104, 9anbi12d 632 . . 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 3427 . 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 34427 . 2 WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
13 df-wlim 34427 . 2 WLim(𝑆, 𝐵) = {𝑥𝐵 ∣ (𝑥 ≠ inf(𝐵, 𝐵, 𝑆) ∧ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))}
1411, 12, 133eqtr4g 2802 1 ((𝑅 = 𝑆𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wne 2944  {crab 3410  Predcpred 6257  supcsup 9383  infcinf 9384  WLimcwlim 34425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-sup 9385  df-inf 9386  df-wlim 34427
This theorem is referenced by:  wlimeq1  34434  wlimeq2  34435
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