| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | limccl.f | . . . . . . . 8
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | 
| 2 | 1 | fdmd 6745 | . . . . . . 7
⊢ (𝜑 → dom 𝐹 = 𝐴) | 
| 3 | 2 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → dom 𝐹 = 𝐴) | 
| 4 |  | limcrcl 25910 | . . . . . . . 8
⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) | 
| 5 | 4 | adantl 481 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) | 
| 6 | 5 | simp2d 1143 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → dom 𝐹 ⊆ ℂ) | 
| 7 | 3, 6 | eqsstrrd 4018 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → 𝐴 ⊆ ℂ) | 
| 8 | 5 | simp3d 1144 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → 𝐵 ∈ ℂ) | 
| 9 | 7, 8 | jca 511 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) | 
| 10 | 9 | ex 412 | . . 3
⊢ (𝜑 → (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ))) | 
| 11 |  | undif1 4475 | . . . . . . 7
⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵}) | 
| 12 |  | difss 4135 | . . . . . . . . . . . 12
⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴 | 
| 13 |  | fssres 6773 | . . . . . . . . . . . 12
⊢ ((𝐹:𝐴⟶ℂ ∧ (𝐴 ∖ {𝐵}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) | 
| 14 | 1, 12, 13 | sylancl 586 | . . . . . . . . . . 11
⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) | 
| 15 | 14 | fdmd 6745 | . . . . . . . . . 10
⊢ (𝜑 → dom (𝐹 ↾ (𝐴 ∖ {𝐵})) = (𝐴 ∖ {𝐵})) | 
| 16 | 15 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → dom (𝐹 ↾ (𝐴 ∖ {𝐵})) = (𝐴 ∖ {𝐵})) | 
| 17 |  | limcrcl 25910 | . . . . . . . . . . 11
⊢ (𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵) → ((𝐹 ↾ (𝐴 ∖ {𝐵})):dom (𝐹 ↾ (𝐴 ∖ {𝐵}))⟶ℂ ∧ dom (𝐹 ↾ (𝐴 ∖ {𝐵})) ⊆ ℂ ∧ 𝐵 ∈ ℂ)) | 
| 18 | 17 | adantl 481 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → ((𝐹 ↾ (𝐴 ∖ {𝐵})):dom (𝐹 ↾ (𝐴 ∖ {𝐵}))⟶ℂ ∧ dom (𝐹 ↾ (𝐴 ∖ {𝐵})) ⊆ ℂ ∧ 𝐵 ∈ ℂ)) | 
| 19 | 18 | simp2d 1143 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → dom (𝐹 ↾ (𝐴 ∖ {𝐵})) ⊆ ℂ) | 
| 20 | 16, 19 | eqsstrrd 4018 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → (𝐴 ∖ {𝐵}) ⊆ ℂ) | 
| 21 | 18 | simp3d 1144 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → 𝐵 ∈ ℂ) | 
| 22 | 21 | snssd 4808 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → {𝐵} ⊆ ℂ) | 
| 23 | 20, 22 | unssd 4191 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ ℂ) | 
| 24 | 11, 23 | eqsstrrid 4022 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → (𝐴 ∪ {𝐵}) ⊆ ℂ) | 
| 25 | 24 | unssad 4192 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → 𝐴 ⊆ ℂ) | 
| 26 | 25, 21 | jca 511 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) | 
| 27 | 26 | ex 412 | . . 3
⊢ (𝜑 → (𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵) → (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ))) | 
| 28 |  | eqid 2736 | . . . . . 6
⊢
((TopOpen‘ℂfld) ↾t (𝐴 ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t (𝐴
∪ {𝐵})) | 
| 29 |  | eqid 2736 | . . . . . 6
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) | 
| 30 |  | eqid 2736 | . . . . . 6
⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) | 
| 31 | 1 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐹:𝐴⟶ℂ) | 
| 32 |  | simprl 770 | . . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ⊆ ℂ) | 
| 33 |  | simprr 772 | . . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ) | 
| 34 | 28, 29, 30, 31, 32, 33 | ellimc 25909 | . . . . 5
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) ∈
((((TopOpen‘ℂfld) ↾t (𝐴 ∪ {𝐵})) CnP
(TopOpen‘ℂfld))‘𝐵))) | 
| 35 | 11 | eqcomi 2745 | . . . . . . 7
⊢ (𝐴 ∪ {𝐵}) = ((𝐴 ∖ {𝐵}) ∪ {𝐵}) | 
| 36 | 35 | oveq2i 7443 | . . . . . 6
⊢
((TopOpen‘ℂfld) ↾t (𝐴 ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t ((𝐴
∖ {𝐵}) ∪ {𝐵})) | 
| 37 | 35 | mpteq1i 5237 | . . . . . . 7
⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) = (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) | 
| 38 |  | elun 4152 | . . . . . . . . 9
⊢ (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 ∈ {𝐵})) | 
| 39 |  | velsn 4641 | . . . . . . . . . . 11
⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵) | 
| 40 | 39 | orbi2i 912 | . . . . . . . . . 10
⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 = 𝐵)) | 
| 41 |  | pm5.61 1002 | . . . . . . . . . . . 12
⊢ (((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ ¬ 𝑧 = 𝐵)) | 
| 42 |  | fvres 6924 | . . . . . . . . . . . . 13
⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧) = (𝐹‘𝑧)) | 
| 43 | 42 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ ¬ 𝑧 = 𝐵) → ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧) = (𝐹‘𝑧)) | 
| 44 | 41, 43 | sylbi 217 | . . . . . . . . . . 11
⊢ (((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) → ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧) = (𝐹‘𝑧)) | 
| 45 | 44 | ifeq2da 4557 | . . . . . . . . . 10
⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 = 𝐵) → if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧)) = if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) | 
| 46 | 40, 45 | sylbi 217 | . . . . . . . . 9
⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 ∈ {𝐵}) → if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧)) = if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) | 
| 47 | 38, 46 | sylbi 217 | . . . . . . . 8
⊢ (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) → if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧)) = if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) | 
| 48 | 47 | mpteq2ia 5244 | . . . . . . 7
⊢ (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧))) = (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) | 
| 49 | 37, 48 | eqtr4i 2767 | . . . . . 6
⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) = (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧))) | 
| 50 | 14 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) | 
| 51 | 32 | ssdifssd 4146 | . . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝐴 ∖ {𝐵}) ⊆ ℂ) | 
| 52 | 36, 29, 49, 50, 51, 33 | ellimc 25909 | . . . . 5
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) ∈
((((TopOpen‘ℂfld) ↾t (𝐴 ∪ {𝐵})) CnP
(TopOpen‘ℂfld))‘𝐵))) | 
| 53 | 34, 52 | bitr4d 282 | . . . 4
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵))) | 
| 54 | 53 | ex 412 | . . 3
⊢ (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)))) | 
| 55 | 10, 27, 54 | pm5.21ndd 379 | . 2
⊢ (𝜑 → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵))) | 
| 56 | 55 | eqrdv 2734 | 1
⊢ (𝜑 → (𝐹 limℂ 𝐵) = ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) |