Theorem smflimmpt 46801

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 1914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
9	7, 8	nfan 1899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
10		smflimmpt.z	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑍 = (ℤ≥‘𝑀)
11	10	uztrn2 12818	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
12	11	adantll 714	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
13		simpll 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
14		smflimmpt.a	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 ∈ 𝑉)
15	14	mptexd 7200	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
16	13, 12, 15	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
17		eqid 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
18	17	fvmpt2 6981	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ 𝑍 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
19	12, 16, 18	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
20	19	dmeqd 5871	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
21		nfv 1914	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥 𝑛 ∈ 𝑍
22	3, 21	nfan 1899	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ 𝑍)
23		nfv 1914	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥 𝑚 ∈ (ℤ≥‘𝑛)
24	22, 23	nfan 1899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛))
25		simplll 774	. . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
30		eqid 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
31	24, 29, 30	fnmptd 6661	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
32	31	fndmd 6625	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
33	20, 32	eqtr2d 2766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝐴 = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
34	9, 33	iineq2d 4981	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 = ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
35	6, 34	iuneq2df 45034	. . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
38	37	eqcomd 2736	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) = ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
39	38	dmeqd 5871	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
40	13, 12, 39	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
41	9, 40	iineq2d 4981	. . . . . . . . . . . . . 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 45034	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵))
44	43	eleq2d 2815	. . . . . . . . . . 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 717	. . . . . . . . 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 4961	. . . . . . . . . . . . 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 717	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴)
51		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝
52	6, 51	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
53		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝
54		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) → (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
55		nfcv 2892	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚𝑥
56		nfii1 4995	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴
57	55, 56	nfel 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴
58	9, 57	nfan 1899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴)
59	10	eluzelz2 45392	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
60	59	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) → 𝑛 ∈ ℤ)
61		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (ℤ≥‘𝑛) = (ℤ≥‘𝑛)
62	10	fvexi 6874	. . . . . . . . . . . . . . . . 17 ⊢ 𝑍 ∈ V
63	62	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) → 𝑍 ∈ V)
64	10	uzssd3 45415	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
65	64	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) → (ℤ≥‘𝑛) ⊆ 𝑍)
66		fvexd 6875	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ V)
67		eliinid 45098	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑥 ∈ 𝐴)
68	67	adantll 714	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑥 ∈ 𝐴)
69	13	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
70	12	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
71	69, 70, 68, 28	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝐵 ∈ 𝑊)
72	30	fvmpt2 6981	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
73	68, 71, 72	syl2anc 584	. . . . . . . . . . . . . . . 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 45680	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) → ((𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ))
75	74	adantllr 719	. . . . . . . . . . . . . 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 716	. . . . . . . . . 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 717	. . . . . . . . 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 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ )
87		simpllr 775	. . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . 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 716	. . . . . . . . . 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 45122	. . . . 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 2765	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ })
99	4	eleq2i 2821	. . . . . . . . 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 3431	. . . . . . . 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 2892	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑥
105		nfiu1 4993	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴
106	51, 105	nfrabw 3446	. . . . . . . . . 10 ⊢ Ⅎ𝑛{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ }
107	4, 106	nfcxfr 2890	. . . . . . . . 9 ⊢ Ⅎ𝑛𝐷
108	104, 107	nfel 2907	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ 𝐷
109	6, 108	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷)
110		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑛( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵))
111	7, 8, 57	nf3an 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴)
112		simp2 1137	. . . . . . . . . . 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 45406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) → (ℤ≥‘𝑛) ⊆ 𝑍)
116		fvexd 6875	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ V)
117	67	3ad2antl3 1188	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑥 ∈ 𝐴)
118		simpl1 1192	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑)
119	112, 11	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
120	118, 119, 117, 28	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝐵 ∈ 𝑊)
121	117, 120, 72	syl2anc 584	. . . . . . . . . 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 45673	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
123	122	3exp 1119	. . . . . . . 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 2736	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))))
128	3, 98, 127	mpteq12da 5192	. . 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 2736	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
130	129	fveq1d 6862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
131	7, 130	mpteq2da 5201	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
132	131	eqcomd 2736	. . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))
133	132	eleq1d 2814	. . . . 5 ⊢ (𝜑 → ((𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ ))
134	3, 42, 133	rabbida2 45119	. . . 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 2736	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))
136	7, 135	mpteq2da 5201	. . . . 5 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))
137	136	fveq2d 6864	. . . 4 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
138	3, 134, 137	mpteq12df 5193	. . 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 2769	. 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 5208	. . 3 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
141		nfcv 2892	. . . 4 ⊢ Ⅎ𝑥𝑍
142		nfmpt1 5208	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
143	141, 142	nfmpt 5207	. . 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 7091	. . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)):𝑍⟶(SMblFn‘𝑆))
148		eqid 2730	. . 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 2730	. . 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 46797	. 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 2829	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  ∃wrex 3054  {crab 3408  Vcvv 3450   ⊆ wss 3916  ∪ ciun 4957  ∩ ciin 4958   ↦ cmpt 5190  dom cdm 5640  ‘cfv 6513  ℤcz 12535  ℤ_≥cuz 12799   ⇝ cli 15456  SAlgcsalg 46299  SMblFncsmblfn 46686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-inf2 9600  ax-cc 10394  ax-ac2 10422  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151  ax-pre-sup 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-iin 4960  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-isom 6522  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-oadd 8440  df-omul 8441  df-er 8673  df-map 8803  df-pm 8804  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-sup 9399  df-inf 9400  df-oi 9469  df-card 9898  df-acn 9901  df-ac 10075  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12188  df-2 12250  df-3 12251  df-n0 12449  df-z 12536  df-uz 12800  df-q 12914  df-rp 12958  df-ioo 13316  df-ico 13318  df-fl 13760  df-seq 13973  df-exp 14033  df-cj 15071  df-re 15072  df-im 15073  df-sqrt 15207  df-abs 15208  df-clim 15460  df-rlim 15461  df-rest 17391  df-salg 46300  df-smblfn 46687

This theorem is referenced by:  smflimsuplem3  46813