Theorem wlimeq12 36011

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 484	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
2		simpl 482	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝑅 = 𝑆)
3	1, 1, 2	infeq123d 9385	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → inf(𝐴, 𝐴, 𝑅) = inf(𝐵, 𝐵, 𝑆))
4	3	neeq2d 2992	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ↔ 𝑥 ≠ inf(𝐵, 𝐵, 𝑆)))
5		equid 2013	. . . . . . 7 ⊢ 𝑥 = 𝑥
6		predeq123 6260	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑥 = 𝑥) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
7	5, 6	mp3an3 1452	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
8	7, 1, 2	supeq123d 9353	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))
9	8	eqeq2d 2747	. . . 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 632	. . 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 3417	. 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 36005	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
13		df-wlim 36005	. 2 ⊢ WLim(𝑆, 𝐵) = {𝑥 ∈ 𝐵 ∣ (𝑥 ≠ inf(𝐵, 𝐵, 𝑆) ∧ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))}
14	11, 12, 13	3eqtr4g 2796	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ≠ wne 2932  {crab 3399  Predcpred 6258  supcsup 9343  infcinf 9344  WLimcwlim 36003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-sup 9345  df-inf 9346  df-wlim 36005

This theorem is referenced by:  wlimeq1  36012  wlimeq2  36013