Theorem climeqmpt 39929

Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
climeqmpt.x	⊢ Ⅎ𝑥𝜑
climeqmpt.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
climeqmpt.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
climeqmpt.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climeqmpt.z	⊢ 𝑍 = (ℤ≥‘𝑀)
climeqmpt.s	⊢ (𝜑 → 𝑍 ⊆ 𝐴)
climeqmpt.t	⊢ (𝜑 → 𝑍 ⊆ 𝐵)
climeqmpt.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈)

Assertion

Ref	Expression
climeqmpt	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑍

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑈(𝑥)   𝑀(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem climeqmpt

Step	Hyp	Ref	Expression
1		climeqmpt.x	. 2 ⊢ Ⅎ𝑥𝜑
2		nfmpt1 4747	. 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
3		nfmpt1 4747	. 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ 𝐶)
4		climeqmpt.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		climeqmpt.z	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
6		climeqmpt.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	6	mptexd 6487	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V)
8		climeqmpt.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9	8	mptexd 6487	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ V)
10		climeqmpt.s	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴)
11	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐴)
12		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝑍)
13	11, 12	sseldd 3604	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐴)
14		climeqmpt.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈)
15		eqid 2622	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
16	15	fvmpt2 6291	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
17	13, 14, 16	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
18		climeqmpt.t	. . . . . . 7 ⊢ (𝜑 → 𝑍 ⊆ 𝐵)
19	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐵)
20	19, 12	sseldd 3604	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐵)
21		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
22	21	fvmpt2 6291	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶)
23	20, 14, 22	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶)
24	23	eqcomd 2628	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥))
25	17, 24	eqtrd 2656	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥))
26	1, 2, 3, 4, 5, 7, 9, 25	climeqf 39920	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  ℤcz 11377  ℤ_≥cuz 11687   ⇝ cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-clim 14219

This theorem is referenced by:  smflimsuplem6  41031  smflimsuplem8  41033