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Theorem ellimc2 25241
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 25239 . . . 4 (𝐹 lim 𝐵) ⊆ ℂ
21sseli 3940 . . 3 (𝐶 ∈ (𝐹 lim 𝐵) → 𝐶 ∈ ℂ)
32pm4.71ri 561 . 2 (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim 𝐵)))
4 eqid 2736 . . . . . 6 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
5 ellimc2.k . . . . . 6 𝐾 = (TopOpen‘ℂfld)
6 eqid 2736 . . . . . 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 25237 . . . . 5 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
1110adantr 481 . . . 4 ((𝜑𝐶 ∈ ℂ) → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
125cnfldtopon 24146 . . . . . . 7 𝐾 ∈ (TopOn‘ℂ)
139snssd 4769 . . . . . . . 8 (𝜑 → {𝐵} ⊆ ℂ)
148, 13unssd 4146 . . . . . . 7 (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
15 resttopon 22512 . . . . . . 7 ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
1612, 14, 15sylancr 587 . . . . . 6 (𝜑 → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
1716adantr 481 . . . . 5 ((𝜑𝐶 ∈ ℂ) → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
1812a1i 11 . . . . 5 ((𝜑𝐶 ∈ ℂ) → 𝐾 ∈ (TopOn‘ℂ))
19 ssun2 4133 . . . . . . 7 {𝐵} ⊆ (𝐴 ∪ {𝐵})
20 snssg 4744 . . . . . . . 8 (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
219, 20syl 17 . . . . . . 7 (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
2219, 21mpbiri 257 . . . . . 6 (𝜑𝐵 ∈ (𝐴 ∪ {𝐵}))
2322adantr 481 . . . . 5 ((𝜑𝐶 ∈ ℂ) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
24 elun 4108 . . . . . . . 8 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧𝐴𝑧 ∈ {𝐵}))
25 velsn 4602 . . . . . . . . 9 (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
2625orbi2i 911 . . . . . . . 8 ((𝑧𝐴𝑧 ∈ {𝐵}) ↔ (𝑧𝐴𝑧 = 𝐵))
2724, 26bitri 274 . . . . . . 7 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧𝐴𝑧 = 𝐵))
28 simpllr 774 . . . . . . . 8 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) ∧ 𝑧 = 𝐵) → 𝐶 ∈ ℂ)
29 pm5.61 999 . . . . . . . . . 10 (((𝑧𝐴𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧𝐴 ∧ ¬ 𝑧 = 𝐵))
307ffvelcdmda 7035 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ ℂ)
3130ad2ant2r 745 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴 ∧ ¬ 𝑧 = 𝐵)) → (𝐹𝑧) ∈ ℂ)
3229, 31sylan2b 594 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ ((𝑧𝐴𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵)) → (𝐹𝑧) ∈ ℂ)
3332anassrs 468 . . . . . . . 8 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) ∧ ¬ 𝑧 = 𝐵) → (𝐹𝑧) ∈ ℂ)
3428, 33ifclda 4521 . . . . . . 7 (((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ ℂ)
3527, 34sylan2b 594 . . . . . 6 (((𝜑𝐶 ∈ ℂ) ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ ℂ)
3635fmpttd 7063 . . . . 5 ((𝜑𝐶 ∈ ℂ) → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))):(𝐴 ∪ {𝐵})⟶ℂ)
37 iscnp 22588 . . . . . 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 540 . . . . 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 1373 . . . 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 4492 . . . . . . . . . . 11 (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = 𝐶)
4140, 6fvmptg 6946 . . . . . . . . . 10 ((𝐵 ∈ (𝐴 ∪ {𝐵}) ∧ 𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) = 𝐶)
4222, 41sylan 580 . . . . . . . . 9 ((𝜑𝐶 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) = 𝐶)
4342eleq1d 2822 . . . . . . . 8 ((𝜑𝐶 ∈ ℂ) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢𝐶𝑢))
4443imbi1d 341 . . . . . . 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 481 . . . . . 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 24147 . . . . . . . . . . 11 𝐾 ∈ Top
47 cnex 11132 . . . . . . . . . . . . . 14 ℂ ∈ V
4847ssex 5278 . . . . . . . . . . . . 13 ((𝐴 ∪ {𝐵}) ⊆ ℂ → (𝐴 ∪ {𝐵}) ∈ V)
4914, 48syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∪ {𝐵}) ∈ V)
5049ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (𝐴 ∪ {𝐵}) ∈ V)
51 restval 17308 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ (𝐴 ∪ {𝐵}) ∈ V) → (𝐾t (𝐴 ∪ {𝐵})) = ran (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
5246, 50, 51sylancr 587 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (𝐾t (𝐴 ∪ {𝐵})) = ran (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
5352rexeqdv 3314 . . . . . . . . 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 3449 . . . . . . . . . . . 12 𝑤 ∈ V
5554inex1 5274 . . . . . . . . . . 11 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
5655rgenw 3068 . . . . . . . . . 10 𝑤𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
57 eqid 2736 . . . . . . . . . . 11 (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))
58 eleq2 2826 . . . . . . . . . . . 12 (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (𝐵𝑣𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
59 imaeq2 6009 . . . . . . . . . . . . 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 3975 . . . . . . . . . . . 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 631 . . . . . . . . . . 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, 61rexrnmptw 7045 . . . . . . . . . 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 728 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
65 elin 3926 . . . . . . . . . . . . 13 (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ (𝐵𝑤𝐵 ∈ (𝐴 ∪ {𝐵})))
6665rbaib 539 . . . . . . . . . . . 12 (𝐵 ∈ (𝐴 ∪ {𝐵}) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵𝑤))
6764, 66syl 17 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵𝑤))
68 simpllr 774 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐶 ∈ ℂ)
69 fvex 6855 . . . . . . . . . . . . . . . . 17 (𝐹𝑧) ∈ V
70 ifexg 4535 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ ℂ ∧ (𝐹𝑧) ∈ V) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ V)
7168, 69, 70sylancl 586 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ V)
7271ralrimivw 3147 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ V)
73 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))
7473fnmpt 6641 . . . . . . . . . . . . . . 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 7058 . . . . . . . . . . . . . . . . 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 6500 . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐶𝑢)
81 elinel2 4156 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑤 ∩ {𝐵}) → 𝑧 ∈ {𝐵})
8225, 40sylbi 216 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ {𝐵} → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = 𝐶)
8382eleq1d 2822 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ {𝐵} → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢𝐶𝑢))
8481, 83syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝑤 ∩ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢𝐶𝑢))
8580, 84syl5ibrcom 246 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝑧 ∈ (𝑤 ∩ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢))
8685ralrimiv 3142 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢)
87 undif1 4435 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵})
8887ineq2i 4169 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = (𝑤 ∩ (𝐴 ∪ {𝐵}))
89 indi 4233 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
9088, 89eqtr3i 2766 . . . . . . . . . . . . . . . . . 18 (𝑤 ∩ (𝐴 ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
9190raleqi 3311 . . . . . . . . . . . . . . . . 17 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢)
92 ralunb 4151 . . . . . . . . . . . . . . . . 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))
9391, 92bitri 274 . . . . . . . . . . . . . . . 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))
9493rbaib 539 . . . . . . . . . . . . . . 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))
9586, 94syl 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))
9679, 95bitr3d 280 . . . . . . . . . . . . 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))
97 elinel2 4156 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
98 eldifsni 4750 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧𝐵)
99 ifnefalse 4498 . . . . . . . . . . . . . . . . 17 (𝑧𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = (𝐹𝑧))
10098, 99syl 17 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝐴 ∖ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = (𝐹𝑧))
101100eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝐴 ∖ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ (𝐹𝑧) ∈ 𝑢))
10297, 101syl 17 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ (𝐹𝑧) ∈ 𝑢))
103102ralbiia 3094 . . . . . . . . . . . . 13 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢)
10496, 103bitrdi 286 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (ran (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢))
105 df-ima 5646 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = ran ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ↾ (𝑤 ∩ (𝐴 ∪ {𝐵})))
106 inss2 4189 . . . . . . . . . . . . . . . 16 (𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵})
107 resmpt 5991 . . . . . . . . . . . . . . . 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))))
108106, 107mp1i 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))))
109108rneqd 5893 . . . . . . . . . . . . . 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))))
110105, 109eqtrid 2788 . . . . . . . . . . . . 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))))
111110sseq1d 3975 . . . . . . . . . . . 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))
1127ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐹:𝐴⟶ℂ)
113112ffund 6672 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → Fun 𝐹)
114 inss2 4189 . . . . . . . . . . . . . . 15 (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
115 difss 4091 . . . . . . . . . . . . . . 15 (𝐴 ∖ {𝐵}) ⊆ 𝐴
116114, 115sstri 3953 . . . . . . . . . . . . . 14 (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ 𝐴
117112fdmd 6679 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → dom 𝐹 = 𝐴)
118116, 117sseqtrrid 3997 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
119 funimass4 6907 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹) → ((𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢))
120113, 118, 119syl2anc 584 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ((𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢))
121104, 111, 1203bitr4d 310 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
12267, 121anbi12d 631 . . . . . . . . . 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)))
123122rexbidva 3173 . . . . . . . . 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)))
12453, 63, 1233bitrd 304 . . . . . . . 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)))
125124anassrs 468 . . . . . . 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)))
126125pm5.74da 802 . . . . . 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))))
12745, 126bitrd 278 . . . . 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))))
128127ralbidva 3172 . . . 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))))
12911, 39, 1283bitrd 304 . . 3 ((𝜑𝐶 ∈ ℂ) → (𝐶 ∈ (𝐹 lim 𝐵) ↔ ∀𝑢𝐾 (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))))
130129pm5.32da 579 . 2 (𝜑 → ((𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim 𝐵)) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢𝐾 (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
1313, 130bitrid 282 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  ifcif 4486  {csn 4586  cmpt 5188  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cc 11049  t crest 17302  TopOpenctopn 17303  fldccnfld 20796  Topctop 22242  TopOnctopon 22259   CnP ccnp 22576   lim climc 25226
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-fz 13425  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-rest 17304  df-topn 17305  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cnp 22579  df-xms 23673  df-ms 23674  df-limc 25230
This theorem is referenced by:  limcnlp  25242  ellimc3  25243  limcflf  25245  limcresi  25249  limciun  25258  lhop1lem  25377  limccog  43851
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