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Theorem ellimc2 24393
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 24391 . . . 4 (𝐹 lim 𝐵) ⊆ ℂ
21sseli 3967 . . 3 (𝐶 ∈ (𝐹 lim 𝐵) → 𝐶 ∈ ℂ)
32pm4.71ri 561 . 2 (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim 𝐵)))
4 eqid 2826 . . . . . 6 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
5 ellimc2.k . . . . . 6 𝐾 = (TopOpen‘ℂfld)
6 eqid 2826 . . . . . 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 24389 . . . . 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 23309 . . . . . . 7 𝐾 ∈ (TopOn‘ℂ)
139snssd 4741 . . . . . . . 8 (𝜑 → {𝐵} ⊆ ℂ)
148, 13unssd 4166 . . . . . . 7 (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
15 resttopon 21688 . . . . . . 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 4153 . . . . . . 7 {𝐵} ⊆ (𝐴 ∪ {𝐵})
20 snssg 4716 . . . . . . . 8 (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
219, 20syl 17 . . . . . . 7 (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
2219, 21mpbiri 259 . . . . . 6 (𝜑𝐵 ∈ (𝐴 ∪ {𝐵}))
2322adantr 481 . . . . 5 ((𝜑𝐶 ∈ ℂ) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
24 elun 4129 . . . . . . . 8 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧𝐴𝑧 ∈ {𝐵}))
25 velsn 4580 . . . . . . . . 9 (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
2625orbi2i 908 . . . . . . . 8 ((𝑧𝐴𝑧 ∈ {𝐵}) ↔ (𝑧𝐴𝑧 = 𝐵))
2724, 26bitri 276 . . . . . . 7 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧𝐴𝑧 = 𝐵))
28 simpllr 772 . . . . . . . 8 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) ∧ 𝑧 = 𝐵) → 𝐶 ∈ ℂ)
29 pm5.61 996 . . . . . . . . . 10 (((𝑧𝐴𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧𝐴 ∧ ¬ 𝑧 = 𝐵))
307ffvelrnda 6847 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ ℂ)
3130ad2ant2r 743 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴 ∧ ¬ 𝑧 = 𝐵)) → (𝐹𝑧) ∈ ℂ)
3229, 31sylan2b 593 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ ((𝑧𝐴𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵)) → (𝐹𝑧) ∈ ℂ)
3332anassrs 468 . . . . . . . 8 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) ∧ ¬ 𝑧 = 𝐵) → (𝐹𝑧) ∈ ℂ)
3428, 33ifclda 4504 . . . . . . 7 (((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ ℂ)
3527, 34sylan2b 593 . . . . . 6 (((𝜑𝐶 ∈ ℂ) ∧ 𝑧 ∈ (𝐴 ∪ {𝐵})) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ ℂ)
3635fmpttd 6875 . . . . 5 ((𝜑𝐶 ∈ ℂ) → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))):(𝐴 ∪ {𝐵})⟶ℂ)
37 iscnp 21764 . . . . . 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 1367 . . . 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 4476 . . . . . . . . . . 11 (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = 𝐶)
4140, 6fvmptg 6763 . . . . . . . . . 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 2902 . . . . . . . 8 ((𝜑𝐶 ∈ ℂ) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))‘𝐵) ∈ 𝑢𝐶𝑢))
4443imbi1d 343 . . . . . . 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 23310 . . . . . . . . . . 11 𝐾 ∈ Top
47 cnex 10607 . . . . . . . . . . . . . 14 ℂ ∈ V
4847ssex 5222 . . . . . . . . . . . . 13 ((𝐴 ∪ {𝐵}) ⊆ ℂ → (𝐴 ∪ {𝐵}) ∈ V)
4914, 48syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∪ {𝐵}) ∈ V)
5049ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (𝐴 ∪ {𝐵}) ∈ V)
51 restval 16690 . . . . . . . . . . 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 3422 . . . . . . . . 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 3503 . . . . . . . . . . . 12 𝑤 ∈ V
5554inex1 5218 . . . . . . . . . . 11 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
5655rgenw 3155 . . . . . . . . . 10 𝑤𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
57 eqid 2826 . . . . . . . . . . 11 (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))
58 eleq2 2906 . . . . . . . . . . . 12 (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (𝐵𝑣𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
59 imaeq2 5923 . . . . . . . . . . . . 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 4002 . . . . . . . . . . . 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 630 . . . . . . . . . . 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 6857 . . . . . . . . . 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 726 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
65 elin 4173 . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐶 ∈ ℂ)
69 fvex 6680 . . . . . . . . . . . . . . . . 17 (𝐹𝑧) ∈ V
70 ifexg 4517 . . . . . . . . . . . . . . . . 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 3188 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ V)
73 eqid 2826 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))
7473fnmpt 6485 . . . . . . . . . . . . . . 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 6870 . . . . . . . . . . . . . . . . 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 6356 . . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐶𝑢)
81 elinel2 4177 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑤 ∩ {𝐵}) → 𝑧 ∈ {𝐵})
8225, 40sylbi 218 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ {𝐵} → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = 𝐶)
8382eleq1d 2902 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ {𝐵} → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢𝐶𝑢))
8481, 83syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝑤 ∩ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢𝐶𝑢))
8580, 84syl5ibrcom 248 . . . . . . . . . . . . . . . 16 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝑧 ∈ (𝑤 ∩ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢))
8685ralrimiv 3186 . . . . . . . . . . . . . . 15 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → ∀𝑧 ∈ (𝑤 ∩ {𝐵})if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢)
87 undif1 4427 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵})
8887ineq2i 4190 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = (𝑤 ∩ (𝐴 ∪ {𝐵}))
89 indi 4254 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
9088, 89eqtr3i 2851 . . . . . . . . . . . . . . . . . 18 (𝑤 ∩ (𝐴 ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
9190raleqi 3419 . . . . . . . . . . . . . . . . 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 4171 . . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . 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 4177 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
98 eldifsni 4721 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧𝐵)
99 ifnefalse 4482 . . . . . . . . . . . . . . . . 17 (𝑧𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = (𝐹𝑧))
10098, 99syl 17 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝐴 ∖ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = (𝐹𝑧))
101100eleq1d 2902 . . . . . . . . . . . . . . 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 3169 . . . . . . . . . . . . 13 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢)
10496, 103syl6bb 288 . . . . . . . . . . . 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 5567 . . . . . . . . . . . . . 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 4210 . . . . . . . . . . . . . . . 16 (𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵})
107 resmpt 5904 . . . . . . . . . . . . . . . 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 5807 . . . . . . . . . . . . . 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, 109syl5eq 2873 . . . . . . . . . . . . 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 4002 . . . . . . . . . . . 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 726 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐹:𝐴⟶ℂ)
113112ffund 6515 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → Fun 𝐹)
114 inss2 4210 . . . . . . . . . . . . . . 15 (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
115 difss 4112 . . . . . . . . . . . . . . 15 (𝐴 ∖ {𝐵}) ⊆ 𝐴
116114, 115sstri 3980 . . . . . . . . . . . . . 14 (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ 𝐴
117112fdmd 6520 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → dom 𝐹 = 𝐴)
118116, 117sseqtrrid 4024 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
119 funimass4 6727 . . . . . . . . . . . . 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 312 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
12267, 121anbi12d 630 . . . . . . . . . 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 3301 . . . . . . . . 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 306 . . . . . . . 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 800 . . . . . 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 280 . . . . 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 3201 . . . 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 306 . . 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, 130syl5bb 284 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 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  Vcvv 3500  cdif 3937  cun 3938  cin 3939  wss 3940  ifcif 4470  {csn 4564  cmpt 5143  dom cdm 5554  ran crn 5555  cres 5556  cima 5557  Fun wfun 6346   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  cc 10524  t crest 16684  TopOpenctopn 16685  fldccnfld 20464  Topctop 21420  TopOnctopon 21437   CnP ccnp 21752   lim climc 24378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fi 8864  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-q 12338  df-rp 12380  df-xneg 12497  df-xadd 12498  df-xmul 12499  df-fz 12883  df-seq 13360  df-exp 13420  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-plusg 16568  df-mulr 16569  df-starv 16570  df-tset 16574  df-ple 16575  df-ds 16577  df-unif 16578  df-rest 16686  df-topn 16687  df-topgen 16707  df-psmet 20456  df-xmet 20457  df-met 20458  df-bl 20459  df-mopn 20460  df-cnfld 20465  df-top 21421  df-topon 21438  df-topsp 21460  df-bases 21473  df-cnp 21755  df-xms 22848  df-ms 22849  df-limc 24382
This theorem is referenced by:  limcnlp  24394  ellimc3  24395  limcflf  24397  limcresi  24401  limciun  24410  lhop1lem  24528  limccog  41769
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