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Theorem ellimc2 23641

Description: Write the definition of a limit directly in terms of open sets of the topology on the complex numbers. (Contributed by Mario Carneiro, 25-Dec-2016.)

**Hypotheses**
Ref	Expression
limccl.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
limccl.a	⊢ (𝜑 → 𝐴 ⊆ ℂ)
limccl.b	⊢ (𝜑 → 𝐵 ∈ ℂ)
ellimc2.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
ellimc2	⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ 𝐾 (𝐶 ∈ 𝑢 → ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))

Distinct variable groups: 𝑤,𝑢,𝐴 𝑢,𝐵,𝑤 𝜑,𝑢,𝑤 𝑢,𝐶,𝑤 𝑢,𝐹,𝑤 𝑢,𝐾,𝑤

Proof of Theorem ellimc2 Dummy variables 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccl 23639	. . . 4 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
2	1	sseli 3599	. . 3 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) → 𝐶 ∈ ℂ)
3	2	pm4.71ri 665	. 2 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim_ℂ 𝐵)))
4		eqid 2622	. . . . . 6 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
5		ellimc2.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
6		eqid 2622	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))
7		limccl.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
8		limccl.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
9		limccl.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
10	4, 5, 6, 7, 8, 9	ellimc 23637	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
11	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
12	5	cnfldtopon 22586	. . . . . . 7 ⊢ 𝐾 ∈ (TopOn‘ℂ)
13	9	snssd 4340	. . . . . . . 8 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
14	8, 13	unssd 3789	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
15		resttopon 20965	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
16	12, 14, 15	sylancr 695	. . . . . 6 ⊢ (𝜑 → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
17	16	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
18	12	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → 𝐾 ∈ (TopOn‘ℂ))
19		ssun2 3777	. . . . . . 7 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
20		snssg 4327	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
21	9, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
22	19, 21	mpbiri 248	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
23	22	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
24		elun 3753	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}))
25		velsn 4193	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
26	25	orbi2i 541	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
27	24, 26	bitri 264	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
28		simpllr 799	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) ∧ 𝑧 = 𝐵) → 𝐶 ∈ ℂ)
29		pm5.61 749	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵))
30	7	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ)
31	30	ad2ant2r 783	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵)) → (𝐹‘𝑧) ∈ ℂ)
32	29, 31	sylan2b 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ ((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵)) → (𝐹‘𝑧) ∈ ℂ)
33	32	anassrs 680	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) ∧ ¬ 𝑧 = 𝐵) → (𝐹‘𝑧) ∈ ℂ)
34	28, 33	ifclda 4120	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
35	27, 34	sylan2b 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
36	35, 6	fmptd 6385	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))):(𝐴 ∪ {𝐵})⟶ℂ)
37		iscnp 21041	. . . . . 6 ⊢ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))):(𝐴 ∪ {𝐵})⟶ℂ ∧ ∀𝑢 ∈ 𝐾 (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢)))))
38	37	baibd 948	. . . . 5 ⊢ ((((𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})) ∧ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))):(𝐴 ∪ {𝐵})⟶ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ ∀𝑢 ∈ 𝐾 (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢))))
39	17, 18, 23, 36, 38	syl31anc 1329	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ ∀𝑢 ∈ 𝐾 (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢))))
40		iftrue 4092	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
41	40, 6	fvmptg 6280	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (𝐴 ∪ {𝐵}) ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
42	22, 41	sylan 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
43	42	eleq1d 2686	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
44	43	imbi1d 331	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → ((((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ (𝐶 ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢))))
45	44	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑢 ∈ 𝐾) → ((((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ (𝐶 ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢))))
46	5	cnfldtop 22587	. . . . . . . . . . 11 ⊢ 𝐾 ∈ Top
47		cnex 10017	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
48	47	ssex 4802	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ {𝐵}) ⊆ ℂ → (𝐴 ∪ {𝐵}) ∈ V)
49	14, 48	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ V)
50	49	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐴 ∪ {𝐵}) ∈ V)
51		restval 16087	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ (𝐴 ∪ {𝐵}) ∈ V) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
52	46, 50, 51	sylancr 695	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
53	52	rexeqdv 3145	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑣 ∈ ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢)))
54		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
55	54	inex1 4799	. . . . . . . . . . 11 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
56	55	rgenw 2924	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
57		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))
58		eleq2 2690	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
59		imaeq2 5462	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) = ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
60	59	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢 ↔ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢))
61	58, 60	anbi12d 747	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → ((𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢) ↔ (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢)))
62	57, 61	rexrnmpt 6369	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ 𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V → (∃𝑣 ∈ ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑤 ∈ 𝐾 (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢)))
63	56, 62	mp1i 13	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (∃𝑣 ∈ ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑤 ∈ 𝐾 (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢)))
64	22	ad3antrrr 766	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
65		elin 3796	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ (𝐵 ∈ 𝑤 ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})))
66	65	rbaib 947	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝐴 ∪ {𝐵}) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵 ∈ 𝑤))
67	64, 66	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵 ∈ 𝑤))
68		simpllr 799	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ ℂ)
69		fvex 6201	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑧) ∈ V
70		ifexg 4157	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ℂ ∧ (𝐹‘𝑧) ∈ V) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
71	68, 69, 70	sylancl 694	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
72	71	ralrimivw 2967	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
73		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))
74	73	fnmpt 6020	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V → (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) Fn (𝑤 ∩ (𝐴 ∪ {𝐵})))
75	73	fmpt 6381	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))):(𝑤 ∩ (𝐴 ∪ {𝐵}))⟶𝑢)
76		df-f 5892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))):(𝑤 ∩ (𝐴 ∪ {𝐵}))⟶𝑢 ↔ ((𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) Fn (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ⊆ 𝑢))
77	75, 76	bitri 264	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ((𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) Fn (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ⊆ 𝑢))
78	77	baib 944	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) Fn (𝑤 ∩ (𝐴 ∪ {𝐵})) → (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ⊆ 𝑢))
79	72, 74, 78	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ⊆ 𝑢))
80		simplrr 801	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ 𝑢)
81		inss2 3834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ {𝐵}) ⊆ {𝐵}
82	81	sseli 3599	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝑤 ∩ {𝐵}) → 𝑧 ∈ {𝐵})
83	25, 40	sylbi 207	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ {𝐵} → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
84	83	eleq1d 2686	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {𝐵} → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
85	82, 84	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝑤 ∩ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
86	80, 85	syl5ibrcom 237	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝑧 ∈ (𝑤 ∩ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢))
87	86	ralrimiv 2965	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢)
88		undif1 4043	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵})
89	88	ineq2i 3811	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = (𝑤 ∩ (𝐴 ∪ {𝐵}))
90		indi 3873	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
91	89, 90	eqtr3i 2646	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
92	91	raleqi 3142	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢)
93		ralunb 3794	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ∧ ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢))
94	92, 93	bitri 264	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ∧ ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢))
95	94	rbaib 947	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 → (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢))
96	87, 95	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢))
97	79, 96	bitr3d 270	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢))
98		inss2 3834	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
99	98	sseli 3599	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
100		eldifsni 4320	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ≠ 𝐵)
101		ifnefalse 4098	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = (𝐹‘𝑧))
102	100, 101	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = (𝐹‘𝑧))
103	102	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ (𝐹‘𝑧) ∈ 𝑢))
104	99, 103	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ (𝐹‘𝑧) ∈ 𝑢))
105	104	ralbiia 2979	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ 𝑢)
106	97, 105	syl6bb 276	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ 𝑢))
107		df-ima 5127	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = ran ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ↾ (𝑤 ∩ (𝐴 ∪ {𝐵})))
108		inss2 3834	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵})
109		resmpt 5449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵}) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ↾ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))))
110	108, 109	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ↾ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))))
111	110	rneqd 5353	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ran ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ↾ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))))
112	107, 111	syl5eq 2668	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))))
113	112	sseq1d 3632	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢 ↔ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) ⊆ 𝑢))
114	7	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐹:𝐴⟶ℂ)
115		ffun 6048	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐴⟶ℂ → Fun 𝐹)
116	114, 115	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → Fun 𝐹)
117		difss 3737	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
118	98, 117	sstri 3612	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ 𝐴
119		fdm 6051	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
120	114, 119	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → dom 𝐹 = 𝐴)
121	118, 120	syl5sseqr 3654	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
122		funimass4 6247	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹) → ((𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ 𝑢))
123	116, 121, 122	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ((𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ 𝑢))
124	106, 113, 123	3bitr4d 300	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
125	67, 124	anbi12d 747	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ((𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢) ↔ (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
126	125	rexbidva 3049	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (∃𝑤 ∈ 𝐾 (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢) ↔ ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
127	53, 63, 126	3bitrd 294	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
128	127	anassrs 680	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑢 ∈ 𝐾) ∧ 𝐶 ∈ 𝑢) → (∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
129	128	pm5.74da 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑢 ∈ 𝐾) → ((𝐶 ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ (𝐶 ∈ 𝑢 → ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
130	45, 129	bitrd 268	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑢 ∈ 𝐾) → ((((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ (𝐶 ∈ 𝑢 → ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
131	130	ralbidva 2985	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (∀𝑢 ∈ 𝐾 (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾 ↾_t (𝐴 ∪ {𝐵}))(𝐵 ∈ 𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ ∀𝑢 ∈ 𝐾 (𝐶 ∈ 𝑢 → ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
132	11, 39, 131	3bitrd 294	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ ∀𝑢 ∈ 𝐾 (𝐶 ∈ 𝑢 → ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
133	132	pm5.32da 673	. 2 ⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim_ℂ 𝐵)) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ 𝐾 (𝐶 ∈ 𝑢 → ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
134	3, 133	syl5bb 272	1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ 𝐾 (𝐶 ∈ 𝑢 → ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ifcif 4086 {csn 4177 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 Topctop 20698 TopOnctopon 20715 CnP ccnp 21029 lim_ℂ climc 23626

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-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

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-rmo 2920 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cnp 21032 df-xms 22125 df-ms 22126 df-limc 23630

This theorem is referenced by: limcnlp 23642 ellimc3 23643 limcflf 23645 limcresi 23649 limciun 23658 lhop1lem 23776 limccog 39852

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