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Theorem ellimc2 23861
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 (𝜑𝐹:𝐴⟶ℂ)
limccl.a (𝜑𝐴 ⊆ ℂ)
limccl.b (𝜑𝐵 ∈ ℂ)
ellimc2.k 𝐾 = (TopOpen‘ℂfld)
Assertion
Ref Expression
ellimc2 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢𝐾 (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (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.
StepHypRef Expression
1 limccl 23859 . . . 4 (𝐹 lim 𝐵) ⊆ ℂ
21sseli 3801 . . 3 (𝐶 ∈ (𝐹 lim 𝐵) → 𝐶 ∈ ℂ)
32pm4.71ri 552 . 2 (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim 𝐵)))
4 eqid 2813 . . . . . 6 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
5 ellimc2.k . . . . . 6 𝐾 = (TopOpen‘ℂfld)
6 eqid 2813 . . . . . 6 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))
7 limccl.f . . . . . 6 (𝜑𝐹:𝐴⟶ℂ)
8 limccl.a . . . . . 6 (𝜑𝐴 ⊆ ℂ)
9 limccl.b . . . . . 6 (𝜑𝐵 ∈ ℂ)
104, 5, 6, 7, 8, 9ellimc 23857 . . . . 5 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
1110adantr 468 . . . 4 ((𝜑𝐶 ∈ ℂ) → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
125cnfldtopon 22803 . . . . . . 7 𝐾 ∈ (TopOn‘ℂ)
139snssd 4537 . . . . . . . 8 (𝜑 → {𝐵} ⊆ ℂ)
148, 13unssd 3995 . . . . . . 7 (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
15 resttopon 21183 . . . . . . 7 ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
1612, 14, 15sylancr 577 . . . . . 6 (𝜑 → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
1716adantr 468 . . . . 5 ((𝜑𝐶 ∈ ℂ) → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
1812a1i 11 . . . . 5 ((𝜑𝐶 ∈ ℂ) → 𝐾 ∈ (TopOn‘ℂ))
19 ssun2 3983 . . . . . . 7 {𝐵} ⊆ (𝐴 ∪ {𝐵})
20 snssg 4512 . . . . . . . 8 (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
219, 20syl 17 . . . . . . 7 (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
2219, 21mpbiri 249 . . . . . 6 (𝜑𝐵 ∈ (𝐴 ∪ {𝐵}))
2322adantr 468 . . . . 5 ((𝜑𝐶 ∈ ℂ) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
24 elun 3959 . . . . . . . 8 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧𝐴𝑧 ∈ {𝐵}))
25 velsn 4393 . . . . . . . . 9 (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
2625orbi2i 927 . . . . . . . 8 ((𝑧𝐴𝑧 ∈ {𝐵}) ↔ (𝑧𝐴𝑧 = 𝐵))
2724, 26bitri 266 . . . . . . 7 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧𝐴𝑧 = 𝐵))
28 simpllr 784 . . . . . . . 8 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) ∧ 𝑧 = 𝐵) → 𝐶 ∈ ℂ)
29 pm5.61 1014 . . . . . . . . . 10 (((𝑧𝐴𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧𝐴 ∧ ¬ 𝑧 = 𝐵))
307ffvelrnda 6584 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ ℂ)
3130ad2ant2r 744 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴 ∧ ¬ 𝑧 = 𝐵)) → (𝐹𝑧) ∈ ℂ)
3229, 31sylan2b 583 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ ((𝑧𝐴𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵)) → (𝐹𝑧) ∈ ℂ)
3332anassrs 455 . . . . . . . 8 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) ∧ ¬ 𝑧 = 𝐵) → (𝐹𝑧) ∈ ℂ)
3428, 33ifclda 4320 . . . . . . 7 (((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ ℂ)
3527, 34sylan2b 583 . . . . . 6 (((𝜑𝐶 ∈ ℂ) ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ ℂ)
3635fmpttd 6610 . . . . 5 ((𝜑𝐶 ∈ ℂ) → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))):(𝐴 ∪ {𝐵})⟶ℂ)
37 iscnp 21259 . . . . . 6 (((𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))):(𝐴 ∪ {𝐵})⟶ℂ ∧ ∀𝑢𝐾 (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢)))))
3837baibd 531 . . . . 5 ((((𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})) ∧ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))):(𝐴 ∪ {𝐵})⟶ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ ∀𝑢𝐾 (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢))))
3917, 18, 23, 36, 38syl31anc 1485 . . . 4 ((𝜑𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵) ↔ ∀𝑢𝐾 (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢))))
40 iftrue 4292 . . . . . . . . . . 11 (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = 𝐶)
4140, 6fvmptg 6504 . . . . . . . . . 10 ((𝐵 ∈ (𝐴 ∪ {𝐵}) ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) = 𝐶)
4222, 41sylan 571 . . . . . . . . 9 ((𝜑𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) = 𝐶)
4342eleq1d 2877 . . . . . . . 8 ((𝜑𝐶 ∈ ℂ) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢𝐶𝑢))
4443imbi1d 332 . . . . . . 7 ((𝜑𝐶 ∈ ℂ) → ((((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ (𝐶𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢))))
4544adantr 468 . . . . . 6 (((𝜑𝐶 ∈ ℂ) ∧ 𝑢𝐾) → ((((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ (𝐶𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢))))
465cnfldtop 22804 . . . . . . . . . . 11 𝐾 ∈ Top
47 cnex 10305 . . . . . . . . . . . . . 14 ℂ ∈ V
4847ssex 5004 . . . . . . . . . . . . 13 ((𝐴 ∪ {𝐵}) ⊆ ℂ → (𝐴 ∪ {𝐵}) ∈ V)
4914, 48syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∪ {𝐵}) ∈ V)
5049ad2antrr 708 . . . . . . . . . . 11 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (𝐴 ∪ {𝐵}) ∈ V)
51 restval 16295 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ (𝐴 ∪ {𝐵}) ∈ V) → (𝐾t (𝐴 ∪ {𝐵})) = ran (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
5246, 50, 51sylancr 577 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (𝐾t (𝐴 ∪ {𝐵})) = ran (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
5352rexeqdv 3341 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑣 ∈ ran (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢)))
54 vex 3401 . . . . . . . . . . . 12 𝑤 ∈ V
5554inex1 5001 . . . . . . . . . . 11 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
5655rgenw 3119 . . . . . . . . . 10 𝑤𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
57 eqid 2813 . . . . . . . . . . 11 (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))
58 eleq2 2881 . . . . . . . . . . . 12 (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (𝐵𝑣𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
59 imaeq2 5679 . . . . . . . . . . . . 13 (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) = ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
6059sseq1d 3836 . . . . . . . . . . . 12 (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢 ↔ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢))
6158, 60anbi12d 618 . . . . . . . . . . 11 (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → ((𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢) ↔ (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢)))
6257, 61rexrnmpt 6594 . . . . . . . . . 10 (∀𝑤𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V → (∃𝑣 ∈ ran (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑤𝐾 (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢)))
6356, 62mp1i 13 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (∃𝑣 ∈ ran (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑤𝐾 (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢)))
6422ad3antrrr 712 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
65 elin 4002 . . . . . . . . . . . . 13 (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ (𝐵𝑤𝐵 ∈ (𝐴 ∪ {𝐵})))
6665rbaib 530 . . . . . . . . . . . 12 (𝐵 ∈ (𝐴 ∪ {𝐵}) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵𝑤))
6764, 66syl 17 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵𝑤))
68 simpllr 784 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐶 ∈ ℂ)
69 fvex 6424 . . . . . . . . . . . . . . . . 17 (𝐹𝑧) ∈ V
70 ifexg 4333 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ℂ ∧ (𝐹𝑧) ∈ V) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ V)
7168, 69, 70sylancl 576 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ V)
7271ralrimivw 3162 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ V)
73 eqid 2813 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))
7473fnmpt 6234 . . . . . . . . . . . . . . 15 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ V → (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) Fn (𝑤 ∩ (𝐴 ∪ {𝐵})))
7573fmpt 6605 . . . . . . . . . . . . . . . . 17 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))):(𝑤 ∩ (𝐴 ∪ {𝐵}))⟶𝑢)
76 df-f 6108 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))):(𝑤 ∩ (𝐴 ∪ {𝐵}))⟶𝑢 ↔ ((𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) Fn (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ⊆ 𝑢))
7775, 76bitri 266 . . . . . . . . . . . . . . . 16 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ((𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) Fn (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ⊆ 𝑢))
7877baib 527 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) Fn (𝑤 ∩ (𝐴 ∪ {𝐵})) → (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ⊆ 𝑢))
7972, 74, 783syl 18 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ⊆ 𝑢))
80 simplrr 787 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐶𝑢)
81 inss2 4037 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∩ {𝐵}) ⊆ {𝐵}
8281sseli 3801 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑤 ∩ {𝐵}) → 𝑧 ∈ {𝐵})
8325, 40sylbi 208 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ {𝐵} → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = 𝐶)
8483eleq1d 2877 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ {𝐵} → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢𝐶𝑢))
8582, 84syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝑤 ∩ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢𝐶𝑢))
8680, 85syl5ibrcom 238 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝑧 ∈ (𝑤 ∩ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢))
8786ralrimiv 3160 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢)
88 undif1 4246 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵})
8988ineq2i 4017 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = (𝑤 ∩ (𝐴 ∪ {𝐵}))
90 indi 4082 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
9189, 90eqtr3i 2837 . . . . . . . . . . . . . . . . . 18 (𝑤 ∩ (𝐴 ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
9291raleqi 3338 . . . . . . . . . . . . . . . . 17 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢)
93 ralunb 4000 . . . . . . . . . . . . . . . . 17 (∀𝑧 ∈ ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ∧ ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢))
9492, 93bitri 266 . . . . . . . . . . . . . . . 16 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ∧ ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢))
9594rbaib 530 . . . . . . . . . . . . . . 15 (∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 → (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢))
9687, 95syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢))
9779, 96bitr3d 272 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢))
98 inss2 4037 . . . . . . . . . . . . . . . 16 (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
9998sseli 3801 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
100 eldifsni 4519 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧𝐵)
101 ifnefalse 4298 . . . . . . . . . . . . . . . . 17 (𝑧𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = (𝐹𝑧))
102100, 101syl 17 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝐴 ∖ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = (𝐹𝑧))
103102eleq1d 2877 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝐴 ∖ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ (𝐹𝑧) ∈ 𝑢))
10499, 103syl 17 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ (𝐹𝑧) ∈ 𝑢))
105104ralbiia 3174 . . . . . . . . . . . . 13 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢)
10697, 105syl6bb 278 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢))
107 df-ima 5331 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = ran ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ↾ (𝑤 ∩ (𝐴 ∪ {𝐵})))
108 inss2 4037 . . . . . . . . . . . . . . . 16 (𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵})
109 resmpt 5661 . . . . . . . . . . . . . . . 16 ((𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵}) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ↾ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))))
110108, 109mp1i 13 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ↾ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))))
111110rneqd 5561 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ran ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ↾ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))))
112107, 111syl5eq 2859 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))))
113112sseq1d 3836 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢 ↔ ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ⊆ 𝑢))
1147ad3antrrr 712 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐹:𝐴⟶ℂ)
115114ffund 6263 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → Fun 𝐹)
116 difss 3943 . . . . . . . . . . . . . . 15 (𝐴 ∖ {𝐵}) ⊆ 𝐴
11798, 116sstri 3814 . . . . . . . . . . . . . 14 (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ 𝐴
118114fdmd 6268 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → dom 𝐹 = 𝐴)
119117, 118syl5sseqr 3858 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
120 funimass4 6471 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹) → ((𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢))
121115, 119, 120syl2anc 575 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ((𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢))
122106, 113, 1213bitr4d 302 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
12367, 122anbi12d 618 . . . . . . . . . 10 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ((𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢) ↔ (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
124123rexbidva 3244 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (∃𝑤𝐾 (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢) ↔ ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
12553, 63, 1243bitrd 296 . . . . . . . 8 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
126125anassrs 455 . . . . . . 7 ((((𝜑𝐶 ∈ ℂ) ∧ 𝑢𝐾) ∧ 𝐶𝑢) → (∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢) ↔ ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
127126pm5.74da 829 . . . . . 6 (((𝜑𝐶 ∈ ℂ) ∧ 𝑢𝐾) → ((𝐶𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
12845, 127bitrd 270 . . . . 5 (((𝜑𝐶 ∈ ℂ) ∧ 𝑢𝐾) → ((((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
129128ralbidva 3180 . . . 4 ((𝜑𝐶 ∈ ℂ) → (∀𝑢𝐾 (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢 → ∃𝑣 ∈ (𝐾t (𝐴 ∪ {𝐵}))(𝐵𝑣 ∧ ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ 𝑣) ⊆ 𝑢)) ↔ ∀𝑢𝐾 (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
13011, 39, 1293bitrd 296 . . 3 ((𝜑𝐶 ∈ ℂ) → (𝐶 ∈ (𝐹 lim 𝐵) ↔ ∀𝑢𝐾 (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
131130pm5.32da 570 . 2 (𝜑 → ((𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim 𝐵)) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢𝐾 (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
1323, 131syl5bb 274 1 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢𝐾 (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2157  wne 2985  wral 3103  wrex 3104  Vcvv 3398  cdif 3773  cun 3774  cin 3775  wss 3776  ifcif 4286  {csn 4377  cmpt 4930  dom cdm 5318  ran crn 5319  cres 5320  cima 5321  Fun wfun 6098   Fn wfn 6099  wf 6100  cfv 6104  (class class class)co 6877  cc 10222  t crest 16289  TopOpenctopn 16290  fldccnfld 19957  Topctop 20915  TopOnctopon 20932   CnP ccnp 21247   lim climc 23846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fi 8559  df-sup 8590  df-inf 8591  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-div 10973  df-nn 11309  df-2 11367  df-3 11368  df-4 11369  df-5 11370  df-6 11371  df-7 11372  df-8 11373  df-9 11374  df-n0 11563  df-z 11647  df-dec 11763  df-uz 11908  df-q 12011  df-rp 12050  df-xneg 12165  df-xadd 12166  df-xmul 12167  df-fz 12553  df-seq 13028  df-exp 13087  df-cj 14065  df-re 14066  df-im 14067  df-sqrt 14201  df-abs 14202  df-struct 16073  df-ndx 16074  df-slot 16075  df-base 16077  df-plusg 16169  df-mulr 16170  df-starv 16171  df-tset 16175  df-ple 16176  df-ds 16178  df-unif 16179  df-rest 16291  df-topn 16292  df-topgen 16312  df-psmet 19949  df-xmet 19950  df-met 19951  df-bl 19952  df-mopn 19953  df-cnfld 19958  df-top 20916  df-topon 20933  df-topsp 20955  df-bases 20968  df-cnp 21250  df-xms 22342  df-ms 22343  df-limc 23850
This theorem is referenced by:  limcnlp  23862  ellimc3  23863  limcflf  23865  limcresi  23869  limciun  23878  lhop1lem  23996  limccog  40333
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