Step | Hyp | Ref
| Expression |
1 | | reloprab 7468 |
. . 3
⊢ Rel
{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} |
2 | 1 | brrelex12i 5732 |
. 2
⊢
(⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V)) |
3 | | df-br 5150 |
. . . . 5
⊢
(⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 ↔ ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}) |
4 | | opex 5465 |
. . . . . . . . 9
⊢
⟨𝑋, 𝑌⟩ ∈ V |
5 | | nfcv 2904 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑤⟨𝑋, 𝑌⟩ |
6 | 5 | nfeq1 2919 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑤⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ |
7 | | nfv 1918 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑤𝜑 |
8 | 6, 7 | nfan 1903 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑤(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
9 | 8 | nfex 2318 |
. . . . . . . . . . 11
⊢
Ⅎ𝑤∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
10 | 9 | nfex 2318 |
. . . . . . . . . 10
⊢
Ⅎ𝑤∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
11 | | nfcv 2904 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑧⟨𝑋, 𝑌⟩ |
12 | 11 | nfeq1 2919 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑧⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ |
13 | | nfsbc1v 3798 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑧[𝑍 / 𝑧]𝜑 |
14 | 12, 13 | nfan 1903 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑧(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) |
15 | 14 | nfex 2318 |
. . . . . . . . . . 11
⊢
Ⅎ𝑧∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) |
16 | 15 | nfex 2318 |
. . . . . . . . . 10
⊢
Ⅎ𝑧∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) |
17 | | eqeq1 2737 |
. . . . . . . . . . . 12
⊢ (𝑤 = ⟨𝑋, 𝑌⟩ → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩)) |
18 | 17 | anbi1d 631 |
. . . . . . . . . . 11
⊢ (𝑤 = ⟨𝑋, 𝑌⟩ → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) |
19 | 18 | 2exbidv 1928 |
. . . . . . . . . 10
⊢ (𝑤 = ⟨𝑋, 𝑌⟩ → (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) |
20 | | sbceq1a 3789 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑍 → (𝜑 ↔ [𝑍 / 𝑧]𝜑)) |
21 | 20 | anbi2d 630 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑍 → ((⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑))) |
22 | 21 | 2exbidv 1928 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑍 → (∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑))) |
23 | 10, 16, 19, 22 | opelopabgf 5541 |
. . . . . . . . 9
⊢
((⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V) →
(⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑))) |
24 | 4, 23 | mpan 689 |
. . . . . . . 8
⊢ (𝑍 ∈ V →
(⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑))) |
25 | | eqcom 2740 |
. . . . . . . . . . . . . . 15
⊢
(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩) |
26 | | vex 3479 |
. . . . . . . . . . . . . . . 16
⊢ 𝑥 ∈ V |
27 | | vex 3479 |
. . . . . . . . . . . . . . . 16
⊢ 𝑦 ∈ V |
28 | 26, 27 | opth 5477 |
. . . . . . . . . . . . . . 15
⊢
(⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ ↔ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) |
29 | 25, 28 | bitri 275 |
. . . . . . . . . . . . . 14
⊢
(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) |
30 | | eqvisset 3492 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑋 → 𝑋 ∈ V) |
31 | | eqvisset 3492 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑌 → 𝑌 ∈ V) |
32 | 30, 31 | anim12i 614 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
33 | 29, 32 | sylbi 216 |
. . . . . . . . . . . . 13
⊢
(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
34 | 33 | adantr 482 |
. . . . . . . . . . . 12
⊢
((⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
35 | 34 | exlimivv 1936 |
. . . . . . . . . . 11
⊢
(∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
36 | 35 | anim1i 616 |
. . . . . . . . . 10
⊢
((∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) ∧ 𝑍 ∈ V) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ V)) |
37 | | df-3an 1090 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ V)) |
38 | 36, 37 | sylibr 233 |
. . . . . . . . 9
⊢
((∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) ∧ 𝑍 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)) |
39 | 38 | expcom 415 |
. . . . . . . 8
⊢ (𝑍 ∈ V → (∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))) |
40 | 24, 39 | sylbid 239 |
. . . . . . 7
⊢ (𝑍 ∈ V →
(⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))) |
41 | 40 | com12 32 |
. . . . . 6
⊢
(⟨⟨𝑋,
𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} → (𝑍 ∈ V → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))) |
42 | | dfoprab2 7467 |
. . . . . 6
⊢
{⟨⟨𝑥,
𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} |
43 | 41, 42 | eleq2s 2852 |
. . . . 5
⊢
(⟨⟨𝑋,
𝑌⟩, 𝑍⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝑍 ∈ V → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))) |
44 | 3, 43 | sylbi 216 |
. . . 4
⊢
(⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑍 ∈ V → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))) |
45 | 44 | com12 32 |
. . 3
⊢ (𝑍 ∈ V → (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))) |
46 | 45 | adantl 483 |
. 2
⊢
((⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V) → (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))) |
47 | 2, 46 | mpcom 38 |
1
⊢
(⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)) |