| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | rexpssxrxp 11306 | . . 3
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) | 
| 2 |  | df-ico 13393 | . . . . 5
⊢ [,) =
(𝑥 ∈
ℝ*, 𝑦
∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | 
| 3 | 2 | ixxf 13397 | . . . 4
⊢
[,):(ℝ* × ℝ*)⟶𝒫
ℝ* | 
| 4 |  | ffn 6736 | . . . 4
⊢
([,):(ℝ* × ℝ*)⟶𝒫
ℝ* → [,) Fn (ℝ* ×
ℝ*)) | 
| 5 |  | fnssresb 6690 | . . . 4
⊢ ([,) Fn
(ℝ* × ℝ*) → (([,) ↾ (ℝ
× ℝ)) Fn (ℝ × ℝ) ↔ (ℝ × ℝ)
⊆ (ℝ* × ℝ*))) | 
| 6 | 3, 4, 5 | mp2b 10 | . . 3
⊢ (([,)
↾ (ℝ × ℝ)) Fn (ℝ × ℝ) ↔ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*)) | 
| 7 | 1, 6 | mpbir 231 | . 2
⊢ ([,)
↾ (ℝ × ℝ)) Fn (ℝ ×
ℝ) | 
| 8 |  | eqid 2737 | . . . . 5
⊢ ([,)
↾ (ℝ × ℝ)) = ([,) ↾ (ℝ ×
ℝ)) | 
| 9 | 8 | icorempo 37352 | . . . 4
⊢ ([,)
↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | 
| 10 | 9 | rneqi 5948 | . . 3
⊢ ran ([,)
↾ (ℝ × ℝ)) = ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | 
| 11 |  | ssrab2 4080 | . . . . . 6
⊢ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ | 
| 12 |  | reex 11246 | . . . . . . 7
⊢ ℝ
∈ V | 
| 13 | 12 | elpw2 5334 | . . . . . 6
⊢ ({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ) | 
| 14 | 11, 13 | mpbir 231 | . . . . 5
⊢ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ | 
| 15 | 14 | rgen2w 3066 | . . . 4
⊢
∀𝑥 ∈
ℝ ∀𝑦 ∈
ℝ {𝑧 ∈ ℝ
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ | 
| 16 |  | eqid 2737 | . . . . . . . 8
⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | 
| 17 | 16 | rnmpo 7566 | . . . . . . 7
⊢ ran
(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) = {𝑙 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}} | 
| 18 | 17 | eqabri 2885 | . . . . . 6
⊢ (𝑙 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | 
| 19 |  | simpl 482 | . . . . . . . . 9
⊢
((∀𝑥 ∈
ℝ ∀𝑦 ∈
ℝ {𝑧 ∈ ℝ
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧
∃𝑥 ∈ ℝ
∃𝑦 ∈ ℝ
𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ) | 
| 20 |  | simpr 484 | . . . . . . . . 9
⊢
((∀𝑥 ∈
ℝ ∀𝑦 ∈
ℝ {𝑧 ∈ ℝ
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧
∃𝑥 ∈ ℝ
∃𝑦 ∈ ℝ
𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | 
| 21 | 19, 20 | r19.29d2r 3140 | . . . . . . . 8
⊢
((∀𝑥 ∈
ℝ ∀𝑦 ∈
ℝ {𝑧 ∈ ℝ
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧
∃𝑥 ∈ ℝ
∃𝑦 ∈ ℝ
𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})) | 
| 22 |  | eleq1 2829 | . . . . . . . . . . 11
⊢ (𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → (𝑙 ∈ 𝒫 ℝ ↔ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫
ℝ)) | 
| 23 | 22 | biimparc 479 | . . . . . . . . . 10
⊢ (({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ) | 
| 24 | 23 | a1i 11 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (({𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ)) | 
| 25 | 24 | rexlimivv 3201 | . . . . . . . 8
⊢
(∃𝑥 ∈
ℝ ∃𝑦 ∈
ℝ ({𝑧 ∈ ℝ
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧ 𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ) | 
| 26 | 21, 25 | syl 17 | . . . . . . 7
⊢
((∀𝑥 ∈
ℝ ∀𝑦 ∈
ℝ {𝑧 ∈ ℝ
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ ∧
∃𝑥 ∈ ℝ
∃𝑦 ∈ ℝ
𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ) | 
| 27 | 26 | ex 412 | . . . . . 6
⊢
(∀𝑥 ∈
ℝ ∀𝑦 ∈
ℝ {𝑧 ∈ ℝ
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ →
(∃𝑥 ∈ ℝ
∃𝑦 ∈ ℝ
𝑙 = {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} → 𝑙 ∈ 𝒫 ℝ)) | 
| 28 | 18, 27 | biimtrid 242 | . . . . 5
⊢
(∀𝑥 ∈
ℝ ∀𝑦 ∈
ℝ {𝑧 ∈ ℝ
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ → (𝑙 ∈ ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) → 𝑙 ∈ 𝒫 ℝ)) | 
| 29 | 28 | ssrdv 3989 | . . . 4
⊢
(∀𝑥 ∈
ℝ ∀𝑦 ∈
ℝ {𝑧 ∈ ℝ
∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ → ran (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ⊆ 𝒫
ℝ) | 
| 30 | 15, 29 | ax-mp 5 | . . 3
⊢ ran
(𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) ⊆ 𝒫
ℝ | 
| 31 | 10, 30 | eqsstri 4030 | . 2
⊢ ran ([,)
↾ (ℝ × ℝ)) ⊆ 𝒫 ℝ | 
| 32 |  | df-f 6565 | . 2
⊢ (([,)
↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫
ℝ ↔ (([,) ↾ (ℝ × ℝ)) Fn (ℝ ×
ℝ) ∧ ran ([,) ↾ (ℝ × ℝ)) ⊆ 𝒫
ℝ)) | 
| 33 | 7, 31, 32 | mpbir2an 711 | 1
⊢ ([,)
↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫
ℝ |