Theorem climeqmpt 41985

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 5166	. 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
3		nfmpt1 5166	. 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 6989	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V)
8		climeqmpt.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9	8	mptexd 6989	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ V)
10		climeqmpt.s	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴)
11	10	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐴)
12		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝑍)
13	11, 12	sseldd 3970	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐴)
14		climeqmpt.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈)
15		eqid 2823	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
16	15	fvmpt2 6781	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
17	13, 14, 16	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
18		climeqmpt.t	. . . . . . 7 ⊢ (𝜑 → 𝑍 ⊆ 𝐵)
19	18	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐵)
20	19, 12	sseldd 3970	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐵)
21		eqid 2823	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
22	21	fvmpt2 6781	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶)
23	20, 14, 22	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶)
24	23	eqcomd 2829	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥))
25	17, 24	eqtrd 2858	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥))
26	1, 2, 3, 4, 5, 7, 9, 25	climeqf 41976	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  Vcvv 3496   ⊆ wss 3938   class class class wbr 5068   ↦ 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-pre-lttri 10613  ax-pre-lttrn 10614

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-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-po 5476  df-so 5477  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-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-neg 10875  df-z 11985  df-uz 12247  df-clim 14847

This theorem is referenced by:  smflimsuplem6  43106  smflimsuplem8  43108