Theorem climfveqmpt3 44384

Description: Two functions that are eventually equal to one another have the same limit. TODO: this is more general than climfveqmpt 44373 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 7222	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V)
4		climfveqmpt3.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
5	4	mptexd 7222	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ V)
6		climfveqmpt3.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		climfveqmpt3.k	. . . . . 6 ⊢ Ⅎ𝑘𝜑
8		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
9	7, 8	nfan 1902	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
10		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑘𝑗
11	10	nfcsb1 3916	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵
12	10	nfcsb1 3916	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐷
13	11, 12	nfeq 2916	. . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷
14	9, 13	nfim 1899	. . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷)
15		eleq1w 2816	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
16	15	anbi2d 629	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
17		csbeq1a 3906	. . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
18		csbeq1a 3906	. . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐷 = ⦋𝑗 / 𝑘⦌𝐷)
19	17, 18	eqeq12d 2748	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵 = 𝐷 ↔ ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷))
20	16, 19	imbi12d 344	. . . 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 2233	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷)
23		climfveqmpt3.i	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴)
24	23	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑍 ⊆ 𝐴)
25		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
26	24, 25	sseldd 3982	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐴)
27		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑘𝑈
28	11, 27	nfel 2917	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈
29	9, 28	nfim 1899	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈)
30	17	eleq1d 2818	. . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ 𝑈 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈))
31	16, 30	imbi12d 344	. . . . 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 2233	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈)
34		eqid 2732	. . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
35	10, 11, 17, 34	fvmptf 7016	. . . 4 ⊢ ((𝑗 ∈ 𝐴 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
36	26, 33, 35	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵)
37		climfveqmpt3.s	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐶)
38	37	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑍 ⊆ 𝐶)
39	38, 25	sseldd 3982	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐶)
40	22, 33	eqeltrrd 2834	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈)
41		eqid 2732	. . . . 5 ⊢ (𝑘 ∈ 𝐶 ↦ 𝐷) = (𝑘 ∈ 𝐶 ↦ 𝐷)
42	10, 12, 18, 41	fvmptf 7016	. . . 4 ⊢ ((𝑗 ∈ 𝐶 ∧ ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷)
43	39, 40, 42	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷)
44	22, 36, 43	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗))
45	1, 3, 5, 6, 44	climfveq 44371	1 ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106  Vcvv 3474  ⦋csb 3892   ⊆ wss 3947   ↦ cmpt 5230  ‘cfv 6540  ℤcz 12554  ℤ_≥cuz 12818   ⇝ cli 15424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428

This theorem is referenced by:  smflimmpt  45512