MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ellimc2 Structured version   Visualization version   GIF version

Theorem ellimc2 25926
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 25924 . . . 4 (𝐹 lim 𝐵) ⊆ ℂ
21sseli 3990 . . 3 (𝐶 ∈ (𝐹 lim 𝐵) → 𝐶 ∈ ℂ)
32pm4.71ri 560 . 2 (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ∈ (𝐹 lim 𝐵)))
4 eqid 2734 . . . . . 6 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
5 ellimc2.k . . . . . 6 𝐾 = (TopOpen‘ℂfld)
6 eqid 2734 . . . . . 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 25922 . . . . 5 (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
1110adantr 480 . . . 4 ((𝜑𝐶 ∈ ℂ) → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
125cnfldtopon 24818 . . . . . . 7 𝐾 ∈ (TopOn‘ℂ)
139snssd 4813 . . . . . . . 8 (𝜑 → {𝐵} ⊆ ℂ)
148, 13unssd 4201 . . . . . . 7 (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
15 resttopon 23184 . . . . . . 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 480 . . . . 5 ((𝜑𝐶 ∈ ℂ) → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
1812a1i 11 . . . . 5 ((𝜑𝐶 ∈ ℂ) → 𝐾 ∈ (TopOn‘ℂ))
19 ssun2 4188 . . . . . . 7 {𝐵} ⊆ (𝐴 ∪ {𝐵})
20 snssg 4787 . . . . . . . 8 (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
219, 20syl 17 . . . . . . 7 (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
2219, 21mpbiri 258 . . . . . 6 (𝜑𝐵 ∈ (𝐴 ∪ {𝐵}))
2322adantr 480 . . . . 5 ((𝜑𝐶 ∈ ℂ) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
24 elun 4162 . . . . . . . 8 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧𝐴𝑧 ∈ {𝐵}))
25 velsn 4646 . . . . . . . . 9 (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
2625orbi2i 912 . . . . . . . 8 ((𝑧𝐴𝑧 ∈ {𝐵}) ↔ (𝑧𝐴𝑧 = 𝐵))
2724, 26bitri 275 . . . . . . 7 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑧𝐴𝑧 = 𝐵))
28 simpllr 776 . . . . . . . 8 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) ∧ 𝑧 = 𝐵) → 𝐶 ∈ ℂ)
29 pm5.61 1002 . . . . . . . . . 10 (((𝑧𝐴𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧𝐴 ∧ ¬ 𝑧 = 𝐵))
307ffvelcdmda 7103 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → (𝐹𝑧) ∈ ℂ)
3130ad2ant2r 747 . . . . . . . . . 10 (((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴 ∧ ¬ 𝑧 = 𝐵)) → (𝐹𝑧) ∈ ℂ)
3229, 31sylan2b 594 . . . . . . . . 9 (((𝜑𝐶 ∈ ℂ) ∧ ((𝑧𝐴𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵)) → (𝐹𝑧) ∈ ℂ)
3332anassrs 467 . . . . . . . 8 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑧𝐴𝑧 = 𝐵)) ∧ ¬ 𝑧 = 𝐵) → (𝐹𝑧) ∈ ℂ)
3428, 33ifclda 4565 . . . . . . 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 7134 . . . . 5 ((𝜑𝐶 ∈ ℂ) → (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))):(𝐴 ∪ {𝐵})⟶ℂ)
37 iscnp 23260 . . . . . 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 539 . . . . 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 1372 . . . 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 4536 . . . . . . . . . . 11 (𝑧 = 𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = 𝐶)
4140, 6fvmptg 7013 . . . . . . . . . 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 2823 . . . . . . . 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 480 . . . . . 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 24819 . . . . . . . . . . 11 𝐾 ∈ Top
47 cnex 11233 . . . . . . . . . . . . . 14 ℂ ∈ V
4847ssex 5326 . . . . . . . . . . . . 13 ((𝐴 ∪ {𝐵}) ⊆ ℂ → (𝐴 ∪ {𝐵}) ∈ V)
4914, 48syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐴 ∪ {𝐵}) ∈ V)
5049ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) → (𝐴 ∪ {𝐵}) ∈ V)
51 restval 17472 . . . . . . . . . . 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 3324 . . . . . . . . 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 3481 . . . . . . . . . . . 12 𝑤 ∈ V
5554inex1 5322 . . . . . . . . . . 11 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
5655rgenw 3062 . . . . . . . . . 10 𝑤𝐾 (𝑤 ∩ (𝐴 ∪ {𝐵})) ∈ V
57 eqid 2734 . . . . . . . . . . 11 (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵}))) = (𝑤𝐾 ↦ (𝑤 ∩ (𝐴 ∪ {𝐵})))
58 eleq2 2827 . . . . . . . . . . . 12 (𝑣 = (𝑤 ∩ (𝐴 ∪ {𝐵})) → (𝐵𝑣𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵}))))
59 imaeq2 6075 . . . . . . . . . . . . 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 4026 . . . . . . . . . . . 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 632 . . . . . . . . . . 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 7114 . . . . . . . . . 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 730 . . . . . . . . . . . 12 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
65 elin 3978 . . . . . . . . . . . . 13 (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ (𝐵𝑤𝐵 ∈ (𝐴 ∪ {𝐵})))
6665rbaib 538 . . . . . . . . . . . 12 (𝐵 ∈ (𝐴 ∪ {𝐵}) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵𝑤))
6764, 66syl 17 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝐵 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↔ 𝐵𝑤))
68 simpllr 776 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐶 ∈ ℂ)
69 fvex 6919 . . . . . . . . . . . . . . . . 17 (𝐹𝑧) ∈ V
70 ifexg 4579 . . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) = (𝑧 ∈ (𝑤 ∩ (𝐴 ∪ {𝐵})) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))
7473fnmpt 6708 . . . . . . . . . . . . . . 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 7129 . . . . . . . . . . . . . . . . 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 6566 . . . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐶𝑢)
81 elinel2 4211 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑤 ∩ {𝐵}) → 𝑧 ∈ {𝐵})
8225, 40sylbi 217 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ {𝐵} → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = 𝐶)
8382eleq1d 2823 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ {𝐵} → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢𝐶𝑢))
8481, 83syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝑤 ∩ {𝐵}) → (if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢𝐶𝑢))
8580, 84syl5ibrcom 247 . . . . . . . . . . . . . . . 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 4481 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵})
8887ineq2i 4224 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = (𝑤 ∩ (𝐴 ∪ {𝐵}))
89 indi 4289 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∩ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
9088, 89eqtr3i 2764 . . . . . . . . . . . . . . . . . 18 (𝑤 ∩ (𝐴 ∪ {𝐵})) = ((𝑤 ∩ (𝐴 ∖ {𝐵})) ∪ (𝑤 ∩ {𝐵}))
9190raleqi 3321 . . . . . . . . . . . . . . . . 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 4206 . . . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . . . . 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 538 . . . . . . . . . . . . . . 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 281 . . . . . . . . . . . . 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 4211 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵})) → 𝑧 ∈ (𝐴 ∖ {𝐵}))
98 eldifsni 4794 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐴 ∖ {𝐵}) → 𝑧𝐵)
99 ifnefalse 4542 . . . . . . . . . . . . . . . . 17 (𝑧𝐵 → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = (𝐹𝑧))
10098, 99syl 17 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝐴 ∖ {𝐵}) → if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) = (𝐹𝑧))
101100eleq1d 2823 . . . . . . . . . . . . . . 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 3088 . . . . . . . . . . . . 13 (∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)) ∈ 𝑢 ↔ ∀𝑧 ∈ (𝑤 ∩ (𝐴 ∖ {𝐵}))(𝐹𝑧) ∈ 𝑢)
10496, 103bitrdi 287 . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . 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 4245 . . . . . . . . . . . . . . . 16 (𝑤 ∩ (𝐴 ∪ {𝐵})) ⊆ (𝐴 ∪ {𝐵})
107 resmpt 6056 . . . . . . . . . . . . . . . 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 5951 . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . 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 4026 . . . . . . . . . . . 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 730 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → 𝐹:𝐴⟶ℂ)
113112ffund 6740 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → Fun 𝐹)
114 inss2 4245 . . . . . . . . . . . . . . 15 (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ (𝐴 ∖ {𝐵})
115 difss 4145 . . . . . . . . . . . . . . 15 (𝐴 ∖ {𝐵}) ⊆ 𝐴
116114, 115sstri 4004 . . . . . . . . . . . . . 14 (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ 𝐴
117112fdmd 6746 . . . . . . . . . . . . . 14 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → dom 𝐹 = 𝐴)
118116, 117sseqtrrid 4048 . . . . . . . . . . . . 13 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (𝑤 ∩ (𝐴 ∖ {𝐵})) ⊆ dom 𝐹)
119 funimass4 6972 . . . . . . . . . . . . 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 311 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ ℂ) ∧ (𝑢𝐾𝐶𝑢)) ∧ 𝑤𝐾) → (((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧))) “ (𝑤 ∩ (𝐴 ∪ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
12267, 121anbi12d 632 . . . . . . . . . 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 3174 . . . . . . . . 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 305 . . . . . . . 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 467 . . . . . . 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 804 . . . . . 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 279 . . . . 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 3173 . . . 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 305 . . 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 283 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 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  Vcvv 3477  cdif 3959  cun 3960  cin 3961  wss 3962  ifcif 4530  {csn 4630  cmpt 5230  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  cc 11150  t crest 17466  TopOpenctopn 17467  fldccnfld 21381  Topctop 22914  TopOnctopon 22931   CnP ccnp 23248   lim climc 25911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fi 9448  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-fz 13544  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-plusg 17310  df-mulr 17311  df-starv 17312  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-rest 17468  df-topn 17469  df-topgen 17489  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cnp 23251  df-xms 24345  df-ms 24346  df-limc 25915
This theorem is referenced by:  limcnlp  25927  ellimc3  25928  limcflf  25930  limcresi  25934  limciun  25943  lhop1lem  26066  limccog  45575
  Copyright terms: Public domain W3C validator