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Theorem ellimc2 25941

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 25939	. . . 4 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
2	1	sseli 3934	. . 3 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) → 𝐶 ∈ ℂ)
3	2	pm4.71ri 568	. 2 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim_ℂ 𝐵)))
4		eqid 2764	. . . . . 6 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
5		ellimc2.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
6		eqid 2764	. . . . . 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 25937	. . . . 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 484	. . . 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 24844	. . . . . . 7 ⊢ 𝐾 ∈ (TopOn‘ℂ)
13	9	snssd 4747	. . . . . . . 8 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
14	8, 13	unssd 4146	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
15		resttopon 23223	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
16	12, 14, 15	sylancr 596	. . . . . 6 ⊢ (𝜑 → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
17	16	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
18	12	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → 𝐾 ∈ (TopOn‘ℂ))
19		ssun2 4133	. . . . . . 7 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
20		snssg 4744	. . . . . . . 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 260	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
23	22	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
24		elun 4108	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}))
25		velsn 4600	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
26	25	orbi2i 923	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
27	24, 26	bitri 277	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵))
28		simpllr 785	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) ∧ 𝑧 = 𝐵) → 𝐶 ∈ ℂ)
29		pm5.61 1014	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵))
30	7	ffvelcdmda 7067	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ)
31	30	ad2ant2r 757	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 = 𝐵)) → (𝐹‘𝑧) ∈ ℂ)
32	29, 31	sylan2b 603	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ ((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵)) → (𝐹‘𝑧) ∈ ℂ)
33	32	anassrs 471	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) ∧ ¬ 𝑧 = 𝐵) → (𝐹‘𝑧) ∈ ℂ)
34	28, 33	ifclda 4518	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐵)) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
35	27, 34	sylan2b 603	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ ℂ)
36	35	fmpttd 7098	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))):(𝐴 ∪ {𝐵})⟶ℂ)
37		iscnp 23299	. . . . . 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 547	. . . . 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 1394	. . . 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 4488	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
41	40, 6	fvmptg 6975	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (𝐴 ∪ {𝐵}) ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
42	22, 41	sylan 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) = 𝐶)
43	42	eleq1d 2849	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ ℂ) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))‘𝐵) ∈ 𝑢 ↔ 𝐶 ∈ 𝑢))
44	43	imbi1d 343	. . . . . . 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 484	. . . . . 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 24845	. . . . . . . . . . 11 ⊢ 𝐾 ∈ Top
47		cnex 11156	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
48	47	ssex 5279	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ {𝐵}) ⊆ ℂ → (𝐴 ∪ {𝐵}) ∈ V)
49	14, 48	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ V)
50	49	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐴 ∪ {𝐵}) ∈ V)
51		restval 17457	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Top ∧ (𝐴 ∪ {𝐵}) ∈ V) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
52	46, 50, 51	sylancr 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ran (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
53	52	rexeqdv 3323	. . . . . . . . 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 3460	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
55	54	inex1 5275	. . . . . . . . . . 11 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
56	55	rgenw 3082	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
57		eqid 2764	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑤 ∈ 𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))
58		eleq2 2853	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (𝐵 ∈ 𝑣 ↔ 𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
59		imaeq2 6047	. . . . . . . . . . . . 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 3969	. . . . . . . . . . . 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 641	. . . . . . . . . . 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 7078	. . . . . . . . . 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 740	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
65		elin 3922	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ (𝐵 ∈ 𝑤 ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})))
66	65	rbaib 546	. . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ ℂ)
69		fvex 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑧) ∈ V
70		ifexg 4532	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ℂ ∧ (𝐹‘𝑧) ∈ V) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
71	68, 69, 70	sylancl 595	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
72	71	ralrimivw 3160	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ V)
73		eqid 2764	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)))
74	73	fnmpt 6663	. . . . . . . . . . . . . . 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 7093	. . . . . . . . . . . . . . . . 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 6527	. . . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . . . 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 543	. . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐶 ∈ 𝑢)
81		elinel2 4156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝑤 ∩ {𝐵}) → 𝑧 ∈ {𝐵})
82	25, 40	sylbi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ {𝐵} → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) = 𝐶)
83	82	eleq1d 2849	. . . . . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝑧 ∈ (𝑤 ∩ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢))
86	85	ralrimiv 3155	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢)
87		undif1 4432	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵})
88	87	ineq2i 4171	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = (𝑤 ∩ (𝐴 ∪ {𝐵}))
89		indi 4238	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
90	88, 89	eqtr3i 2789	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
91	90	raleqi 3320	. . . . . . . . . . . . . . . . 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 4151	. . . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . . . 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 546	. . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . 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 4156	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
98		eldifsni 4752	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧 ≠ 𝐵)
99		ifnefalse 4494	. . . . . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . 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 3108	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹‘𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹‘𝑧) ∈ 𝑢)
104	96, 103	bitrdi 289	. . . . . . . . . . . 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 5662	. . . . . . . . . . . . . 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 4191	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵})
107		resmpt 6028	. . . . . . . . . . . . . . . 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 5916	. . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . 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 3969	. . . . . . . . . . . 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 740	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → 𝐹:𝐴⟶ℂ)
113	112	ffund 6698	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → Fun 𝐹)
114		inss2 4191	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
115		difss 4091	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
116	114, 115	sstri 3947	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ 𝐴
117	112	fdmd 6704	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → dom 𝐹 = 𝐴)
118	116, 117	sseqtrrid 3981	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐶 ∈ ℂ) ∧ (𝑢 ∈ 𝐾 ∧ 𝐶 ∈ 𝑢)) ∧ 𝑤 ∈ 𝐾) → (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
119		funimass4 6933	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 313	. . . . . . . . . . 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 641	. . . . . . . . . 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 3186	. . . . . . . . 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 307	. . . . . . . 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 471	. . . . . . 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 813	. . . . . 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 281	. . . . 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 3185	. . . 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 307	. . 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 587	. 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 285	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 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1099 = wceq 1562 ∈ wcel 2144 ≠ wne 2959 ∀wral 3078 ∃wrex 3088 Vcvv 3456 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ifcif 4482 {csn 4584 ↦ cmpt 5183 dom cdm 5649 ran crn 5650 ↾ cres 5651 “ cima 5652 Fun wfun 6517 Fn wfn 6518 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 ℂcc 11073 ↾_t crest 17451 TopOpenctopn 17452 ℂ_fldccnfld 21426 Topctop 22955 TopOnctopon 22972 CnP ccnp 23287 lim_ℂ climc 25926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-er 8680 df-map 8812 df-pm 8813 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fi 9359 df-sup 9390 df-inf 9391 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-fz 13515 df-seq 14017 df-exp 14077 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17248 df-plusg 17301 df-mulr 17302 df-starv 17303 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-rest 17453 df-topn 17454 df-topgen 17474 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cnp 23290 df-xms 24382 df-ms 24383 df-limc 25930

This theorem is referenced by: limcnlp 25942 ellimc3 25943 limcflf 25945 limcresi 25949 limciun 25958 lhop1lem 26077 limccog 46201

W3C validator