Theorem wlimeq12 36207

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 489	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
2		simpl 487	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝑅 = 𝑆)
3	1, 1, 2	infeq123d 9441	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → inf(𝐴, 𝐴, 𝑅) = inf(𝐵, 𝐵, 𝑆))
4	3	neeq2d 3024	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ↔ 𝑥 ≠ inf(𝐵, 𝐵, 𝑆)))
5		equid 2039	. . . . . . 7 ⊢ 𝑥 = 𝑥
6		predeq123 6304	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑥 = 𝑥) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
7	5, 6	mp3an3 1476	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
8	7, 1, 2	supeq123d 9409	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))
9	8	eqeq2d 2780	. . . 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 643	. . 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 3441	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))} = {𝑥 ∈ 𝐵 ∣ (𝑥 ≠ inf(𝐵, 𝐵, 𝑆) ∧ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))})
12		df-wlim 36201	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
13		df-wlim 36201	. 2 ⊢ WLim(𝑆, 𝐵) = {𝑥 ∈ 𝐵 ∣ (𝑥 ≠ inf(𝐵, 𝐵, 𝑆) ∧ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))}
14	11, 12, 13	3eqtr4g 2829	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ≠ wne 2964  {crab 3423  Predcpred 6302  supcsup 9399  infcinf 9400  WLimcwlim 36199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-sup 9401  df-inf 9402  df-wlim 36201

This theorem is referenced by:  wlimeq1  36208  wlimeq2  36209