Theorem wlimeq12 33813

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 485	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
2		simpl 483	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → 𝑅 = 𝑆)
3	1, 1, 2	infeq123d 9240	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → inf(𝐴, 𝐴, 𝑅) = inf(𝐵, 𝐵, 𝑆))
4	3	neeq2d 3004	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ↔ 𝑥 ≠ inf(𝐵, 𝐵, 𝑆)))
5		equid 2015	. . . . . . 7 ⊢ 𝑥 = 𝑥
6		predeq123 6203	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑥 = 𝑥) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
7	5, 6	mp3an3 1449	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑆, 𝐵, 𝑥))
8	7, 1, 2	supeq123d 9209	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅) = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))
9	8	eqeq2d 2749	. . . 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 631	. . 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 3420	. 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 33807	. 2 ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
13		df-wlim 33807	. 2 ⊢ WLim(𝑆, 𝐵) = {𝑥 ∈ 𝐵 ∣ (𝑥 ≠ inf(𝐵, 𝐵, 𝑆) ∧ 𝑥 = sup(Pred(𝑆, 𝐵, 𝑥), 𝐵, 𝑆))}
14	11, 12, 13	3eqtr4g 2803	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ≠ wne 2943  {crab 3068  Predcpred 6201  supcsup 9199  infcinf 9200  WLimcwlim 33805

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-sup 9201  df-inf 9202  df-wlim 33807

This theorem is referenced by:  wlimeq1  33814  wlimeq2  33815