| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | indistop 23009 | . 2
⊢ {∅,
𝐴} ∈
Top | 
| 2 |  | simpl 482 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → 𝑥 ∈ ∪ {∅, 𝐴}) | 
| 3 |  | 0ex 5307 | . . . . . . . . . . . 12
⊢ ∅
∈ V | 
| 4 |  | n0i 4340 | . . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ∪ {∅, 𝐴} → ¬ ∪
{∅, 𝐴} =
∅) | 
| 5 |  | prprc2 4766 | . . . . . . . . . . . . . . . 16
⊢ (¬
𝐴 ∈ V → {∅,
𝐴} =
{∅}) | 
| 6 | 5 | unieqd 4920 | . . . . . . . . . . . . . . 15
⊢ (¬
𝐴 ∈ V → ∪ {∅, 𝐴} = ∪
{∅}) | 
| 7 | 3 | unisn 4926 | . . . . . . . . . . . . . . 15
⊢ ∪ {∅} = ∅ | 
| 8 | 6, 7 | eqtrdi 2793 | . . . . . . . . . . . . . 14
⊢ (¬
𝐴 ∈ V → ∪ {∅, 𝐴} = ∅) | 
| 9 | 4, 8 | nsyl2 141 | . . . . . . . . . . . . 13
⊢ (𝑥 ∈ ∪ {∅, 𝐴} → 𝐴 ∈ V) | 
| 10 | 9 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → 𝐴 ∈ V) | 
| 11 |  | uniprg 4923 | . . . . . . . . . . . 12
⊢ ((∅
∈ V ∧ 𝐴 ∈ V)
→ ∪ {∅, 𝐴} = (∅ ∪ 𝐴)) | 
| 12 | 3, 10, 11 | sylancr 587 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → ∪ {∅, 𝐴} = (∅ ∪ 𝐴)) | 
| 13 |  | uncom 4158 | . . . . . . . . . . . 12
⊢ (∅
∪ 𝐴) = (𝐴 ∪ ∅) | 
| 14 |  | un0 4394 | . . . . . . . . . . . 12
⊢ (𝐴 ∪ ∅) = 𝐴 | 
| 15 | 13, 14 | eqtri 2765 | . . . . . . . . . . 11
⊢ (∅
∪ 𝐴) = 𝐴 | 
| 16 | 12, 15 | eqtrdi 2793 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → ∪ {∅, 𝐴} = 𝐴) | 
| 17 | 2, 16 | eleqtrd 2843 | . . . . . . . . 9
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → 𝑥 ∈ 𝐴) | 
| 18 |  | simpr 484 | . . . . . . . . . 10
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → 𝑦 ∈ ∪ {∅, 𝐴}) | 
| 19 | 18, 16 | eleqtrd 2843 | . . . . . . . . 9
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → 𝑦 ∈ 𝐴) | 
| 20 | 17, 19 | ifcld 4572 | . . . . . . . 8
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) →
if(𝑧 = 0, 𝑥, 𝑦) ∈ 𝐴) | 
| 21 | 20 | adantr 480 | . . . . . . 7
⊢ (((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) ∧ 𝑧 ∈ (0[,]1)) → if(𝑧 = 0, 𝑥, 𝑦) ∈ 𝐴) | 
| 22 | 21 | fmpttd 7135 | . . . . . 6
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)):(0[,]1)⟶𝐴) | 
| 23 |  | ovex 7464 | . . . . . . 7
⊢ (0[,]1)
∈ V | 
| 24 |  | elmapg 8879 | . . . . . . 7
⊢ ((𝐴 ∈ V ∧ (0[,]1) ∈
V) → ((𝑧 ∈
(0[,]1) ↦ if(𝑧 = 0,
𝑥, 𝑦)) ∈ (𝐴 ↑m (0[,]1)) ↔ (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)):(0[,]1)⟶𝐴)) | 
| 25 | 10, 23, 24 | sylancl 586 | . . . . . 6
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) →
((𝑧 ∈ (0[,]1) ↦
if(𝑧 = 0, 𝑥, 𝑦)) ∈ (𝐴 ↑m (0[,]1)) ↔ (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)):(0[,]1)⟶𝐴)) | 
| 26 | 22, 25 | mpbird 257 | . . . . 5
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (𝐴 ↑m
(0[,]1))) | 
| 27 |  | iitopon 24905 | . . . . . 6
⊢ II ∈
(TopOn‘(0[,]1)) | 
| 28 |  | cnindis 23300 | . . . . . 6
⊢ ((II
∈ (TopOn‘(0[,]1)) ∧ 𝐴 ∈ V) → (II Cn {∅, 𝐴}) = (𝐴 ↑m
(0[,]1))) | 
| 29 | 27, 10, 28 | sylancr 587 | . . . . 5
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → (II Cn
{∅, 𝐴}) = (𝐴 ↑m
(0[,]1))) | 
| 30 | 26, 29 | eleqtrrd 2844 | . . . 4
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) → (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (II Cn {∅, 𝐴})) | 
| 31 |  | 0elunit 13509 | . . . . 5
⊢ 0 ∈
(0[,]1) | 
| 32 |  | iftrue 4531 | . . . . . 6
⊢ (𝑧 = 0 → if(𝑧 = 0, 𝑥, 𝑦) = 𝑥) | 
| 33 |  | eqid 2737 | . . . . . 6
⊢ (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) | 
| 34 |  | vex 3484 | . . . . . 6
⊢ 𝑥 ∈ V | 
| 35 | 32, 33, 34 | fvmpt 7016 | . . . . 5
⊢ (0 ∈
(0[,]1) → ((𝑧 ∈
(0[,]1) ↦ if(𝑧 = 0,
𝑥, 𝑦))‘0) = 𝑥) | 
| 36 | 31, 35 | mp1i 13 | . . . 4
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) →
((𝑧 ∈ (0[,]1) ↦
if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥) | 
| 37 |  | 1elunit 13510 | . . . . 5
⊢ 1 ∈
(0[,]1) | 
| 38 |  | ax-1ne0 11224 | . . . . . . . 8
⊢ 1 ≠
0 | 
| 39 |  | neeq1 3003 | . . . . . . . 8
⊢ (𝑧 = 1 → (𝑧 ≠ 0 ↔ 1 ≠ 0)) | 
| 40 | 38, 39 | mpbiri 258 | . . . . . . 7
⊢ (𝑧 = 1 → 𝑧 ≠ 0) | 
| 41 |  | ifnefalse 4537 | . . . . . . 7
⊢ (𝑧 ≠ 0 → if(𝑧 = 0, 𝑥, 𝑦) = 𝑦) | 
| 42 | 40, 41 | syl 17 | . . . . . 6
⊢ (𝑧 = 1 → if(𝑧 = 0, 𝑥, 𝑦) = 𝑦) | 
| 43 |  | vex 3484 | . . . . . 6
⊢ 𝑦 ∈ V | 
| 44 | 42, 33, 43 | fvmpt 7016 | . . . . 5
⊢ (1 ∈
(0[,]1) → ((𝑧 ∈
(0[,]1) ↦ if(𝑧 = 0,
𝑥, 𝑦))‘1) = 𝑦) | 
| 45 | 37, 44 | mp1i 13 | . . . 4
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) →
((𝑧 ∈ (0[,]1) ↦
if(𝑧 = 0, 𝑥, 𝑦))‘1) = 𝑦) | 
| 46 |  | fveq1 6905 | . . . . . . 7
⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0)) | 
| 47 | 46 | eqeq1d 2739 | . . . . . 6
⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → ((𝑓‘0) = 𝑥 ↔ ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥)) | 
| 48 |  | fveq1 6905 | . . . . . . 7
⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1)) | 
| 49 | 48 | eqeq1d 2739 | . . . . . 6
⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → ((𝑓‘1) = 𝑦 ↔ ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1) = 𝑦)) | 
| 50 | 47, 49 | anbi12d 632 | . . . . 5
⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ (((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1) = 𝑦))) | 
| 51 | 50 | rspcev 3622 | . . . 4
⊢ (((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (II Cn {∅, 𝐴}) ∧ (((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1) = 𝑦)) → ∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)) | 
| 52 | 30, 36, 45, 51 | syl12anc 837 | . . 3
⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪
{∅, 𝐴}) →
∃𝑓 ∈ (II Cn
{∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)) | 
| 53 | 52 | rgen2 3199 | . 2
⊢
∀𝑥 ∈
∪ {∅, 𝐴}∀𝑦 ∈ ∪
{∅, 𝐴}∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) | 
| 54 |  | eqid 2737 | . . 3
⊢ ∪ {∅, 𝐴} = ∪ {∅,
𝐴} | 
| 55 | 54 | ispconn 35228 | . 2
⊢
({∅, 𝐴} ∈
PConn ↔ ({∅, 𝐴}
∈ Top ∧ ∀𝑥
∈ ∪ {∅, 𝐴}∀𝑦 ∈ ∪
{∅, 𝐴}∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) | 
| 56 | 1, 53, 55 | mpbir2an 711 | 1
⊢ {∅,
𝐴} ∈
PConn |