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Theorem ellimc2 26045

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 26043	. . . 4 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
2	1	sseli 3933	. . 3 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) → 𝐶 ∈ ℂ)
3	2	pm4.71ri 569	. 2 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim_ℂ 𝐵)))
4		eqid 2763	. . . . . 6 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
5		ellimc2.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
6		eqid 2763	. . . . . 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 26041	. . . . 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 485	. . . 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 24948	. . . . . . 7 ⊢ 𝐾 ∈ (TopOn‘ℂ)
13	9	snssd 4752	. . . . . . . 8 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
14	8, 13	unssd 4145	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
15		resttopon 23327	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
16	12, 14, 15	sylancr 598	. . . . . 6 ⊢ (𝜑 → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
17	16	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
18	12	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → 𝐾 ∈ (TopOn‘ℂ))
19		ssun2 4132	. . . . . . 7 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
20		snssg 4749	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
21	9, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
22	19, 21	mpbiri 261	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
23	22	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
24		elun 4107	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}))
25		velsn 4605	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
26	25	orbi2i 925	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
27	24, 26	bitri 278	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
28		simpllr 787	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) ∧ 𝑧 = 𝐵) → 𝐶 ∈ ℂ)
29		pm5.61 1016	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵))
30	7	ffvelcdmda 7079	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ)
31	30	ad2ant2r 759	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵)) → (𝐹‘𝑧) ∈ ℂ)
32	29, 31	sylan2b 605	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ ((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵)) → (𝐹‘𝑧) ∈ ℂ)
33	32	anassrs 472	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) ∧ ¬ 𝑧 = 𝐵) → (𝐹‘𝑧) ∈ ℂ)
34	28, 33	ifclda 4523	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
35	27, 34	sylan2b 605	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
36	35	fmpttd 7110	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))):(𝐴 ∪ {𝐵})⟶ℂ)
37		iscnp 23403	. . . . . 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 548	. . . . 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 1400	. . . 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 4493	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
41	40, 6	fvmptg 6987	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (𝐴 ∪ {𝐵}) ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
42	22, 41	sylan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
43	42	eleq1d 2848	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
44	43	imbi1d 344	. . . . . . 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 485	. . . . . 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 24949	. . . . . . . . . . 11 ⊢ 𝐾 ∈ Top
47		cnex 11185	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
48	47	ssex 5291	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ {𝐵}) ⊆ ℂ → (𝐴 ∪ {𝐵}) ∈ V)
49	14, 48	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ V)
50	49	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐴 ∪ {𝐵}) ∈ V)
51		restval 17483	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ (𝐴 ∪ {𝐵}) ∈ V) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
52	46, 50, 51	sylancr 598	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
53	52	rexeqdv 3324	. . . . . . . . 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 3459	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
55	54	inex1 5286	. . . . . . . . . . 11 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
56	55	rgenw 3083	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
57		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))
58		eleq2 2852	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
59		imaeq2 6058	. . . . . . . . . . . . 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 3968	. . . . . . . . . . . 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 643	. . . . . . . . . . 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	rexrnmptw 7090	. . . . . . . . . 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 14	. . . . . . . . 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 742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
65		elin 3921	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ (𝐵 ∈ 𝑤 ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})))
66	65	rbaib 547	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝐴 ∪ {𝐵}) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵 ∈ 𝑤))
67	64, 66	syl 18	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵 ∈ 𝑤))
68		simpllr 787	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ ℂ)
69		fvex 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑧) ∈ V
70		ifexg 4537	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ℂ ∧ (𝐹‘𝑧) ∈ V) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
71	68, 69, 70	sylancl 597	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
72	71	ralrimivw 3161	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
73		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))
74	73	fnmpt 6675	. . . . . . . . . . . . . . 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 7105	. . . . . . . . . . . . . . . . 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 6540	. . . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . . 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 544	. . . . . . . . . . . . . . 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 19	. . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ 𝑢)
81		elinel2 4155	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝑤 ∩ {𝐵}) → 𝑧 ∈ {𝐵})
82	25, 40	sylbi 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ {𝐵} → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
83	82	eleq1d 2848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {𝐵} → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
84	81, 83	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝑤 ∩ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
85	80, 84	syl5ibrcom 250	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝑧 ∈ (𝑤 ∩ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢))
86	85	ralrimiv 3156	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢)
87		undif1 4437	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵})
88	87	ineq2i 4170	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = (𝑤 ∩ (𝐴 ∪ {𝐵}))
89		indi 4237	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
90	88, 89	eqtr3i 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
91	90	raleqi 3321	. . . . . . . . . . . . . . . . 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)
92		ralunb 4150	. . . . . . . . . . . . . . . . 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))
93	91, 92	bitri 278	. . . . . . . . . . . . . . . 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))
94	93	rbaib 547	. . . . . . . . . . . . . . 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))
95	86, 94	syl 18	. . . . . . . . . . . . . 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))
96	79, 95	bitr3d 284	. . . . . . . . . . . . 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))
97		elinel2 4155	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
98		eldifsni 4758	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ≠ 𝐵)
99		ifnefalse 4499	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = (𝐹‘𝑧))
100	98, 99	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = (𝐹‘𝑧))
101	100	eleq1d 2848	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ (𝐹‘𝑧) ∈ 𝑢))
102	97, 101	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ (𝐹‘𝑧) ∈ 𝑢))
103	102	ralbiia 3109	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ 𝑢)
104	96, 103	bitrdi 290	. . . . . . . . . . . 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))
105		df-ima 5674	. . . . . . . . . . . . . 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})))
106		inss2 4190	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵})
107		resmpt 6039	. . . . . . . . . . . . . . . 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))))
108	106, 107	mp1i 14	. . . . . . . . . . . . . . 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))))
109	108	rneqd 5928	. . . . . . . . . . . . . 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))))
110	105, 109	eqtrid 2810	. . . . . . . . . . . . 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))))
111	110	sseq1d 3968	. . . . . . . . . . . 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))
112	7	ad3antrrr 742	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐹:𝐴⟶ℂ)
113	112	ffund 6710	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → Fun 𝐹)
114		inss2 4190	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
115		difss 4090	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
116	114, 115	sstri 3946	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ 𝐴
117	112	fdmd 6716	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → dom 𝐹 = 𝐴)
118	116, 117	sseqtrrid 3980	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
119		funimass4 6945	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹) → ((𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ 𝑢))
120	113, 118, 119	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ((𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ 𝑢))
121	104, 111, 120	3bitr4d 314	. . . . . . . . . . 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))
122	67, 121	anbi12d 643	. . . . . . . . . 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)))
123	122	rexbidva 3187	. . . . . . . . 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)))
124	53, 63, 123	3bitrd 308	. . . . . . . 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)))
125	124	anassrs 472	. . . . . . 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)))
126	125	pm5.74da 815	. . . . . 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))))
127	45, 126	bitrd 282	. . . . 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))))
128	127	ralbidva 3186	. . . 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))))
129	11, 39, 128	3bitrd 308	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ ∀𝑢 ∈ 𝐾 (𝐶 ∈ 𝑢 → ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
130	129	pm5.32da 589	. 2 ⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim_ℂ 𝐵)) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢 ∈ 𝐾 (𝐶 ∈ 𝑢 → ∃𝑤 ∈ 𝐾 (𝐵 ∈ 𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
131	3, 130	bitrid 286	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ∖ cdif 3902 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ifcif 4487 {csn 4589 ↦ cmpt 5192 dom cdm 5661 ran crn 5662 ↾ cres 5663 “ cima 5664 Fun wfun 6530 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11102 ↾_t crest 17477 TopOpenctopn 17478 ℂ_fldccnfld 21531 Topctop 23059 TopOnctopon 23076 CnP ccnp 23391 lim_ℂ climc 26030

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9367 df-sup 9398 df-inf 9399 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-fz 13540 df-seq 14043 df-exp 14103 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-struct 17211 df-slot 17246 df-ndx 17258 df-base 17274 df-plusg 17327 df-mulr 17328 df-starv 17329 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-rest 17479 df-topn 17480 df-topgen 17500 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-cnfld 21532 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-cnp 23394 df-xms 24486 df-ms 24487 df-limc 26034

This theorem is used by: limcnlp 26046 ellimc3 26047 limcflf 26049 limcresi 26053 limciun 26062 lhop1lem 26181 limccog 46364

W3C validator