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Theorem ellimc2 25776

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 25774	. . . 4 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
2	1	sseli 3931	. . 3 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) → 𝐶 ∈ ℂ)
3	2	pm4.71ri 560	. 2 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim_ℂ 𝐵)))
4		eqid 2729	. . . . . 6 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
5		ellimc2.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
6		eqid 2729	. . . . . 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 25772	. . . . 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 480	. . . 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 24668	. . . . . . 7 ⊢ 𝐾 ∈ (TopOn‘ℂ)
13	9	snssd 4760	. . . . . . . 8 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
14	8, 13	unssd 4143	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
15		resttopon 23046	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
16	12, 14, 15	sylancr 587	. . . . . 6 ⊢ (𝜑 → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
17	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
18	12	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → 𝐾 ∈ (TopOn‘ℂ))
19		ssun2 4130	. . . . . . 7 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
20		snssg 4735	. . . . . . . 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 258	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
24		elun 4104	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}))
25		velsn 4593	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
26	25	orbi2i 912	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
27	24, 26	bitri 275	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
28		simpllr 775	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) ∧ 𝑧 = 𝐵) → 𝐶 ∈ ℂ)
29		pm5.61 1002	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵))
30	7	ffvelcdmda 7018	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ)
31	30	ad2ant2r 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵)) → (𝐹‘𝑧) ∈ ℂ)
32	29, 31	sylan2b 594	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ ((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵)) → (𝐹‘𝑧) ∈ ℂ)
33	32	anassrs 467	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) ∧ ¬ 𝑧 = 𝐵) → (𝐹‘𝑧) ∈ ℂ)
34	28, 33	ifclda 4512	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
35	27, 34	sylan2b 594	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
36	35	fmpttd 7049	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))):(𝐴 ∪ {𝐵})⟶ℂ)
37		iscnp 23122	. . . . . 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 539	. . . . 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 1375	. . . 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 4482	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
41	40, 6	fvmptg 6928	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (𝐴 ∪ {𝐵}) ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
42	22, 41	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
43	42	eleq1d 2813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
44	43	imbi1d 341	. . . . . . 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 480	. . . . . 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 24669	. . . . . . . . . . 11 ⊢ 𝐾 ∈ Top
47		cnex 11090	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
48	47	ssex 5260	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ {𝐵}) ⊆ ℂ → (𝐴 ∪ {𝐵}) ∈ V)
49	14, 48	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ V)
50	49	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐴 ∪ {𝐵}) ∈ V)
51		restval 17330	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ (𝐴 ∪ {𝐵}) ∈ V) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
52	46, 50, 51	sylancr 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
53	52	rexeqdv 3290	. . . . . . . . 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 3440	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
55	54	inex1 5256	. . . . . . . . . . 11 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
56	55	rgenw 3048	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
57		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))
58		eleq2 2817	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
59		imaeq2 6007	. . . . . . . . . . . . 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 3967	. . . . . . . . . . . 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 632	. . . . . . . . . . 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 7029	. . . . . . . . . 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 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
65		elin 3919	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ (𝐵 ∈ 𝑤 ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})))
66	65	rbaib 538	. . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ ℂ)
69		fvex 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑧) ∈ V
70		ifexg 4526	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ℂ ∧ (𝐹‘𝑧) ∈ V) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
71	68, 69, 70	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
72	71	ralrimivw 3125	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
73		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))
74	73	fnmpt 6622	. . . . . . . . . . . . . . 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 7044	. . . . . . . . . . . . . . . . 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 6486	. . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . 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 535	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ 𝑢)
81		elinel2 4153	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝑤 ∩ {𝐵}) → 𝑧 ∈ {𝐵})
82	25, 40	sylbi 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ {𝐵} → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
83	82	eleq1d 2813	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {𝐵} → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
84	81, 83	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝑤 ∩ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
85	80, 84	syl5ibrcom 247	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝑧 ∈ (𝑤 ∩ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢))
86	85	ralrimiv 3120	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢)
87		undif1 4427	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵})
88	87	ineq2i 4168	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = (𝑤 ∩ (𝐴 ∪ {𝐵}))
89		indi 4235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
90	88, 89	eqtr3i 2754	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
91	90	raleqi 3287	. . . . . . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . 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 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))
96	79, 95	bitr3d 281	. . . . . . . . . . . . 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 4153	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
98		eldifsni 4741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ≠ 𝐵)
99		ifnefalse 4488	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = (𝐹‘𝑧))
100	98, 99	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = (𝐹‘𝑧))
101	100	eleq1d 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ (𝐹‘𝑧) ∈ 𝑢))
102	97, 101	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → (if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ (𝐹‘𝑧) ∈ 𝑢))
103	102	ralbiia 3073	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ 𝑢)
104	96, 103	bitrdi 287	. . . . . . . . . . . 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 5632	. . . . . . . . . . . . . 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 4189	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵})
107		resmpt 5988	. . . . . . . . . . . . . . . 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 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))))
109	108	rneqd 5880	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 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 3967	. . . . . . . . . . . 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 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐹:𝐴⟶ℂ)
113	112	ffund 6656	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → Fun 𝐹)
114		inss2 4189	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
115		difss 4087	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
116	114, 115	sstri 3945	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ 𝐴
117	112	fdmd 6662	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → dom 𝐹 = 𝐴)
118	116, 117	sseqtrrid 3979	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
119		funimass4 6887	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 311	. . . . . . . . . . 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 632	. . . . . . . . . 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 3151	. . . . . . . . 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 305	. . . . . . . 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 467	. . . . . . 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 803	. . . . . 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 279	. . . . 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 3150	. . . 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 305	. . 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 579	. 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 283	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3436 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ifcif 4476 {csn 4577 ↦ cmpt 5173 dom cdm 5619 ran crn 5620 ↾ cres 5621 “ cima 5622 Fun wfun 6476 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ↾_t crest 17324 TopOpenctopn 17325 ℂ_fldccnfld 21261 Topctop 22778 TopOnctopon 22795 CnP ccnp 23110 lim_ℂ climc 25761

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fi 9301 df-sup 9332 df-inf 9333 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-fz 13411 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-plusg 17174 df-mulr 17175 df-starv 17176 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-rest 17326 df-topn 17327 df-topgen 17347 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cnp 23113 df-xms 24206 df-ms 24207 df-limc 25765

This theorem is referenced by: limcnlp 25777 ellimc3 25778 limcflf 25780 limcresi 25784 limciun 25793 lhop1lem 25916 limccog 45611

W3C validator