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Theorem ellimc2 25858

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 25856	. . . 4 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
2	1	sseli 3918	. . 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 2737	. . . . . 6 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
5		ellimc2.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
6		eqid 2737	. . . . . 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 25854	. . . . 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 24761	. . . . . . 7 ⊢ 𝐾 ∈ (TopOn‘ℂ)
13	9	snssd 4753	. . . . . . . 8 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
14	8, 13	unssd 4133	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
15		resttopon 23140	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
16	12, 14, 15	sylancr 588	. . . . . 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 4120	. . . . . . 7 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
20		snssg 4728	. . . . . . . 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 4094	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}))
25		velsn 4584	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
26	25	orbi2i 913	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
27	24, 26	bitri 275	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
28		simpllr 776	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) ∧ 𝑧 = 𝐵) → 𝐶 ∈ ℂ)
29		pm5.61 1003	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵))
30	7	ffvelcdmda 7032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ)
31	30	ad2ant2r 748	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵)) → (𝐹‘𝑧) ∈ ℂ)
32	29, 31	sylan2b 595	. . . . . . . . 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 4503	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
35	27, 34	sylan2b 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
36	35	fmpttd 7063	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))):(𝐴 ∪ {𝐵})⟶ℂ)
37		iscnp 23216	. . . . . 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 1376	. . . 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 4473	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
41	40, 6	fvmptg 6941	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (𝐴 ∪ {𝐵}) ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
42	22, 41	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
43	42	eleq1d 2822	. . . . . . . 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 24762	. . . . . . . . . . 11 ⊢ 𝐾 ∈ Top
47		cnex 11114	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
48	47	ssex 5259	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ {𝐵}) ⊆ ℂ → (𝐴 ∪ {𝐵}) ∈ V)
49	14, 48	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ V)
50	49	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐴 ∪ {𝐵}) ∈ V)
51		restval 17384	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ (𝐴 ∪ {𝐵}) ∈ V) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
52	46, 50, 51	sylancr 588	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
53	52	rexeqdv 3297	. . . . . . . . 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 3434	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
55	54	inex1 5255	. . . . . . . . . . 11 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
56	55	rgenw 3056	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
57		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))
58		eleq2 2826	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
59		imaeq2 6017	. . . . . . . . . . . . 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 3954	. . . . . . . . . . . 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 633	. . . . . . . . . . 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 7043	. . . . . . . . . 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 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
65		elin 3906	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ ℂ)
69		fvex 6849	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑧) ∈ V
70		ifexg 4517	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ℂ ∧ (𝐹‘𝑧) ∈ V) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
71	68, 69, 70	sylancl 587	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
72	71	ralrimivw 3134	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
73		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))
74	73	fnmpt 6634	. . . . . . . . . . . . . . 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 7058	. . . . . . . . . . . . . . . . 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 6498	. . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ 𝑢)
81		elinel2 4143	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝑤 ∩ {𝐵}) → 𝑧 ∈ {𝐵})
82	25, 40	sylbi 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ {𝐵} → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
83	82	eleq1d 2822	. . . . . . . . . . . . . . . . . 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 3129	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢)
87		undif1 4417	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵})
88	87	ineq2i 4158	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = (𝑤 ∩ (𝐴 ∪ {𝐵}))
89		indi 4225	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
90	88, 89	eqtr3i 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
91	90	raleqi 3294	. . . . . . . . . . . . . . . . 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 4138	. . . . . . . . . . . . . . . . 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 4143	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
98		eldifsni 4734	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ≠ 𝐵)
99		ifnefalse 4479	. . . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . 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 3082	. . . . . . . . . . . . 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 5639	. . . . . . . . . . . . . 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 4179	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵})
107		resmpt 5998	. . . . . . . . . . . . . . . 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 5889	. . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . 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 3954	. . . . . . . . . . . 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 731	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐹:𝐴⟶ℂ)
113	112	ffund 6668	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → Fun 𝐹)
114		inss2 4179	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
115		difss 4077	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
116	114, 115	sstri 3932	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ 𝐴
117	112	fdmd 6674	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → dom 𝐹 = 𝐴)
118	116, 117	sseqtrrid 3966	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
119		funimass4 6900	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 633	. . . . . . . . . 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 3160	. . . . . . . . 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 804	. . . . . 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 3159	. . . 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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 Vcvv 3430 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ifcif 4467 {csn 4568 ↦ cmpt 5167 dom cdm 5626 ran crn 5627 ↾ cres 5628 “ cima 5629 Fun wfun 6488 Fn wfn 6489 ⟶wf 6490 ‘cfv 6494 (class class class)co 7362 ℂcc 11031 ↾_t crest 17378 TopOpenctopn 17379 ℂ_fldccnfld 21348 Topctop 22872 TopOnctopon 22889 CnP ccnp 23204 lim_ℂ climc 25843

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

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-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-map 8770 df-pm 8771 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fi 9319 df-sup 9350 df-inf 9351 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-fz 13457 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-plusg 17228 df-mulr 17229 df-starv 17230 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-rest 17380 df-topn 17381 df-topgen 17401 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-cnfld 21349 df-top 22873 df-topon 22890 df-topsp 22912 df-bases 22925 df-cnp 23207 df-xms 24299 df-ms 24300 df-limc 25847

This theorem is referenced by: limcnlp 25859 ellimc3 25860 limcflf 25862 limcresi 25866 limciun 25875 lhop1lem 25994 limccog 46072

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