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Theorem wlimeq12 33003
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 485 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵) → 𝐴 = 𝐵)
2 simpl 483 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵) → 𝑅 = 𝑆)
31, 1, 2infeq123d 8933 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵) → inf(𝐴, 𝐴, 𝑅) = inf(𝐵, 𝐵, 𝑆))
43neeq2d 3073 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ↔ 𝑥 ≠ inf(𝐵, 𝐵, 𝑆)))
5 equid 2010 . . . . . . 7 𝑥 = 𝑥
6 predeq123 6142 . . . . . . 7 ((𝑅 = 𝑆𝐴 = 𝐵𝑥 = 𝑥) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
75, 6mp3an3 1441 . . . . . 6 ((𝑅 = 𝑆𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
87, 1, 2supeq123d 8902 . . . . 5 ((𝑅 = 𝑆𝐴 = 𝐵) → sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))
98eqeq2d 2829 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) ↔ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆)))
104, 9anbi12d 630 . . 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 3483 . 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 32997 . 2 WLim(𝑅, 𝐴) = {𝑥𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
13 df-wlim 32997 . 2 WLim(𝑆, 𝐵) = {𝑥𝐵 ∣ (𝑥 ≠ inf(𝐵, 𝐵, 𝑆) ∧ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))}
1411, 12, 133eqtr4g 2878 1 ((𝑅 = 𝑆𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wne 3013  {crab 3139  Predcpred 6140  supcsup 8892  infcinf 8893  WLimcwlim 32995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-sup 8894  df-inf 8895  df-wlim 32997
This theorem is referenced by:  wlimeq1  33004  wlimeq2  33005
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