smflimmpt - Mathbox for Glauco Siliprandi

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Theorem smflimmpt 47242

Description: The limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). 𝐴 can contain 𝑚 as a free variable, in other words it can be thought as an indexed collection 𝐴(𝑚). 𝐵 can be thought as a collection with two indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
smflimmpt.p	⊢ Ⅎ𝑚𝜑
smflimmpt.x	⊢ Ⅎ𝑥𝜑
smflimmpt.n	⊢ Ⅎ𝑛𝜑
smflimmpt.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimmpt.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
smflimmpt.a	⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 ∈ 𝑉)
smflimmpt.b	⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
smflimmpt.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimmpt.l	⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
smflimmpt.d	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ }
smflimmpt.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵)))

**Assertion**
Ref	Expression
smflimmpt	⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Distinct variable groups: 𝐴,𝑛,𝑥 𝐵,𝑛 𝑆,𝑚,𝑛 𝑚,𝑍,𝑛,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑚,𝑛) 𝐴(𝑚) 𝐵(𝑥,𝑚) 𝐷(𝑥,𝑚,𝑛) 𝑆(𝑥) 𝐺(𝑥,𝑚,𝑛) 𝑀(𝑥,𝑚,𝑛) 𝑉(𝑥,𝑚,𝑛) 𝑊(𝑥,𝑚,𝑛)

**Proof of Theorem smflimmpt**
Step	Hyp	Ref	Expression
1		smflimmpt.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵))))
3		smflimmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
4		smflimmpt.d	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ }
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ })
6		smflimmpt.n	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛𝜑
7		smflimmpt.p	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝜑
8		nfv 1916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
9	7, 8	nfan 1901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
10		smflimmpt.z	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑍 = (ℤ_≥‘𝑀)
11	10	uztrn2 12771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
12	11	adantll 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
13		simpll 767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
14		smflimmpt.a	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 ∈ 𝑉)
15	14	mptexd 7170	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
16	13, 12, 15	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
17		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
18	17	fvmpt2 6951	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ 𝑍 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
19	12, 16, 18	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
20	19	dmeqd 5852	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
21		nfv 1916	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥 𝑛 ∈ 𝑍
22	3, 21	nfan 1901	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ 𝑍)
23		nfv 1916	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥 𝑚 ∈ (ℤ_≥‘𝑛)
24	22, 23	nfan 1901	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))
25		simplll 775	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) ∧ 𝑥 ∈ 𝐴) → 𝜑)
26	12	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) ∧ 𝑥 ∈ 𝐴) → 𝑚 ∈ 𝑍)
27		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
28		smflimmpt.b	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
29	25, 26, 27, 28	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
30		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
31	24, 29, 30	fnmptd 6631	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
32	31	fndmd 6595	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
33	20, 32	eqtr2d 2773	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐴 = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
34	9, 33	iineq2d 4958	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 = ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
35	6, 34	iuneq2df 45481	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
36		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
37	36, 15, 18	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
38	37	eqcomd 2743	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) = ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
39	38	dmeqd 5852	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
40	13, 12, 39	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
41	9, 40	iineq2d 4958	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
42	6, 41	iuneq2df 45481	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
43	35, 42	eqtr4d 2775	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵))
44	43	eleq2d 2823	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ↔ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵)))
45	44	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵))
46	45	adantrr 718	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵))
47		eliun 4938	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
48	47	biimpi 216	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
49	48	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
50	49	adantrr 718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
51		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝
52	6, 51	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
53		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝
54		simpllr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
55		nfcv 2899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚𝑥
56		nfii1 4972	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴
57	55, 56	nfel 2914	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴
58	9, 57	nfan 1901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
59	10	eluzelz2 45835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
60	59	ad2antlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑛 ∈ ℤ)
61		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
62	10	fvexi 6846	. . . . . . . . . . . . . . . . 17 ⊢ 𝑍 ∈ V
63	62	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑍 ∈ V)
64	10	uzssd3 45858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
65	64	ad2antlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (ℤ_≥‘𝑛) ⊆ 𝑍)
66		fvexd 6847	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ V)
67		eliinid 45544	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ 𝐴)
68	67	adantll 715	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ 𝐴)
69	13	adantlr 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
70	12	adantlr 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
71	69, 70, 68, 28	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐵 ∈ 𝑊)
72	30	fvmpt2 6951	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
73	68, 71, 72	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
74	58, 60, 61, 63, 63, 65, 65, 66, 73	climeldmeqmpt3 46121	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → ((𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ))
75	74	adantllr 720	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → ((𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ))
76	54, 75	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )
77	76	exp31 419	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )))
78	52, 53, 77	rexlimd 3245	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ))
79	78	adantrl 717	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ))
80	50, 79	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )) → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )
81	46, 80	jca 511	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ))
82	81	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )))
83	44	biimpar 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
84	83	adantrr 718	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
85	84, 48	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
86	6, 53	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )
87		simpllr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )
88	74	adantllr 720	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → ((𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ))
89	87, 88	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
90	89	exp31 419	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )))
91	86, 51, 90	rexlimd 3245	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ))
92	91	adantrl 717	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ))
93	85, 92	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )) → (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
94	84, 93	jca 511	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ))
95	94	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )))
96	82, 95	impbid 212	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) ↔ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )))
97	3, 96	rabbida3 45568	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ } = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ })
98	5, 97	eqtrd 2772	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ })
99	4	eleq2i 2829	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ })
100	99	biimpi 216	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ })
101		rabidim1 3412	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ } → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
102	100, 101, 48	3syl 18	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
103	102	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
104		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑥
105		nfiu1 4970	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴
106	51, 105	nfrabw 3427	. . . . . . . . . 10 ⊢ Ⅎ𝑛{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ }
107	4, 106	nfcxfr 2897	. . . . . . . . 9 ⊢ Ⅎ𝑛𝐷
108	104, 107	nfel 2914	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ 𝐷
109	6, 108	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
110		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑛( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵))
111	7, 8, 57	nf3an 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
112		simp2 1138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑛 ∈ 𝑍)
113	112, 59	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑛 ∈ ℤ)
114	62	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑍 ∈ V)
115	10, 112	uzssd2 45849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (ℤ_≥‘𝑛) ⊆ 𝑍)
116		fvexd 6847	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ V)
117	67	3ad2antl3 1189	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ 𝐴)
118		simpl1 1193	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
119	112, 11	sylan 581	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
120	118, 119, 117, 28	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐵 ∈ 𝑊)
121	117, 120, 72	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
122	111, 113, 61, 114, 114, 115, 115, 116, 121	climfveqmpt3 46114	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
123	122	3exp 1120	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵)))))
124	123	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵)))))
125	109, 110, 124	rexlimd 3245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵))))
126	103, 125	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
127	126	eqcomd 2743	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))))
128	3, 98, 127	mpteq12da 5169	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))))
129	38	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
130	129	fveq1d 6834	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
131	7, 130	mpteq2da 5178	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
132	131	eqcomd 2743	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))
133	132	eleq1d 2822	. . . . 5 ⊢ (𝜑 → ((𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ ))
134	3, 42, 133	rabbida2 45565	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ } = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ })
135	130	eqcomd 2743	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))
136	7, 135	mpteq2da 5178	. . . . 5 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))
137	136	fveq2d 6836	. . . 4 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
138	3, 134, 137	mpteq12df 5170	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))))
139	2, 128, 138	3eqtrd 2776	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))))
140		nfmpt1 5185	. . 3 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
141		nfcv 2899	. . . 4 ⊢ Ⅎ𝑥𝑍
142		nfmpt1 5185	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
143	141, 142	nfmpt 5184	. . 3 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
144		smflimmpt.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
145		smflimmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
146		smflimmpt.l	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
147	7, 146, 17	fmptdf 7061	. . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)):𝑍⟶(SMblFn‘𝑆))
148		eqid 2737	. . 3 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ }
149		eqid 2737	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
150	140, 143, 144, 10, 145, 147, 148, 149	smflim2 47238	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
151	139, 150	eqeltrd 2837	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 Ⅎwnf 1785 ∈ wcel 2114 ∃wrex 3062 {crab 3390 Vcvv 3430 ⊆ wss 3890 ∪ ciun 4934 ∩ ciin 4935 ↦ cmpt 5167 dom cdm 5622 ‘cfv 6490 ℤcz 12489 ℤ_≥cuz 12752 ⇝ cli 15408 SAlgcsalg 46740 SMblFncsmblfn 47127

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cc 10346 ax-ac2 10374 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-oadd 8400 df-omul 8401 df-er 8634 df-map 8766 df-pm 8767 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-acn 9855 df-ac 10027 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12753 df-q 12863 df-rp 12907 df-ioo 13266 df-ico 13268 df-fl 13713 df-seq 13926 df-exp 13986 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-clim 15412 df-rlim 15413 df-rest 17343 df-salg 46741 df-smblfn 47128

This theorem is referenced by: smflimsuplem3 47254

W3C validator