Theorem climfveqmpt3 41970

Description: Two functions that are eventually equal to one another have the same limit. TODO: this is more general than climfveqmpt 41959 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
climfveqmpt3.k	⊢ Ⅎ𝑘𝜑
climfveqmpt3.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climfveqmpt3.z	⊢ 𝑍 = (ℤ≥‘𝑀)
climfveqmpt3.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
climfveqmpt3.c	⊢ (𝜑 → 𝐶 ∈ 𝑊)
climfveqmpt3.i	⊢ (𝜑 → 𝑍 ⊆ 𝐴)
climfveqmpt3.s	⊢ (𝜑 → 𝑍 ⊆ 𝐶)
climfveqmpt3.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈)
climfveqmpt3.d	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷)

Assertion

Ref	Expression
climfveqmpt3	⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷)))

Distinct variable groups:   𝐴,𝑘   𝐶,𝑘   𝑈,𝑘   𝑘,𝑍

Allowed substitution hints:   𝜑(𝑘)   𝐵(𝑘)   𝐷(𝑘)   𝑀(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem climfveqmpt3

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climfveqmpt3.z	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		climfveqmpt3.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3	2	mptexd 6989	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
4		climfveqmpt3.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
5	4	mptexd 6989	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ V)
6		climfveqmpt3.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		climfveqmpt3.k	. . . . . 6 ⊢ Ⅎ𝑘𝜑
8		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
9	7, 8	nfan 1900	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
10		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑘𝑗
11	10	nfcsb1 3908	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵
12	10	nfcsb1 3908	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐷
13	11, 12	nfeq 2993	. . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷
14	9, 13	nfim 1897	. . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷)
15		eleq1w 2897	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
16	15	anbi2d 630	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
17		csbeq1a 3899	. . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
18		csbeq1a 3899	. . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐷 = ⦋𝑗 / 𝑘⦌𝐷)
19	17, 18	eqeq12d 2839	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵 = 𝐷 ↔ ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷))
20	16, 19	imbi12d 347	. . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷)))
21		climfveqmpt3.d	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷)
22	14, 20, 21	chvarfv 2242	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷)
23		climfveqmpt3.i	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴)
24	23	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑍 ⊆ 𝐴)
25		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
26	24, 25	sseldd 3970	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐴)
27		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑘𝑈
28	11, 27	nfel 2994	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈
29	9, 28	nfim 1897	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈)
30	17	eleq1d 2899	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ 𝑈 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈))
31	16, 30	imbi12d 347	. . . . 5 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈)))
32		climfveqmpt3.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈)
33	29, 31, 32	chvarfv 2242	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈)
34		eqid 2823	. . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
35	10, 11, 17, 34	fvmptf 6791	. . . 4 ⊢ ((𝑗 ∈ 𝐴 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
36	26, 33, 35	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
37		climfveqmpt3.s	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐶)
38	37	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑍 ⊆ 𝐶)
39	38, 25	sseldd 3970	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐶)
40	22, 33	eqeltrrd 2916	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈)
41		eqid 2823	. . . . 5 ⊢ (𝑘 ∈ 𝐶 ↦ 𝐷) = (𝑘 ∈ 𝐶 ↦ 𝐷)
42	10, 12, 18, 41	fvmptf 6791	. . . 4 ⊢ ((𝑗 ∈ 𝐶 ∧ ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷)
43	39, 40, 42	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷)
44	22, 36, 43	3eqtr4d 2868	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗))
45	1, 3, 5, 6, 44	climfveq 41957	1 ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  Vcvv 3496  ⦋csb 3885   ⊆ wss 3938   ↦ cmpt 5148  ‘cfv 6357  ℤcz 11984  ℤ_≥cuz 12246   ⇝ cli 14843

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-sup 8908  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-seq 13373  df-exp 13433  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847

This theorem is referenced by:  smflimmpt  43091