Step | Hyp | Ref
| Expression |
1 | | df-cross 5765 |
. . 3
⊢ Cross = (x ∈ V, y ∈ V ↦ (x × y)) |
2 | | rexcom 2773 |
. . . . 5
⊢ (∃a ∈ x ∃b ∈ y z = ⟨a, b⟩ ↔ ∃b ∈ y ∃a ∈ x z = ⟨a, b⟩) |
3 | | elxp2 4803 |
. . . . 5
⊢ (z ∈ (x × y)
↔ ∃a ∈ x ∃b ∈ y z = ⟨a, b⟩) |
4 | | elin 3220 |
. . . . . . . 8
⊢ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) ↔ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins2 Ins2 S ∧ ⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c))) |
5 | | snex 4112 |
. . . . . . . . . . 11
⊢ {z} ∈
V |
6 | 5 | otelins2 5792 |
. . . . . . . . . 10
⊢ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{b}, ⟨x, y⟩⟩ ∈ Ins2 S ) |
7 | | vex 2863 |
. . . . . . . . . . 11
⊢ x ∈
V |
8 | 7 | otelins2 5792 |
. . . . . . . . . 10
⊢ (⟨{b}, ⟨x, y⟩⟩ ∈ Ins2 S ↔ ⟨{b}, y⟩ ∈ S
) |
9 | | vex 2863 |
. . . . . . . . . . 11
⊢ b ∈
V |
10 | | vex 2863 |
. . . . . . . . . . 11
⊢ y ∈
V |
11 | 9, 10 | opelssetsn 4761 |
. . . . . . . . . 10
⊢ (⟨{b}, y⟩ ∈ S ↔ b ∈ y) |
12 | 6, 8, 11 | 3bitri 262 |
. . . . . . . . 9
⊢ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins2 Ins2 S ↔ b ∈ y) |
13 | 10 | oqelins4 5795 |
. . . . . . . . . 10
⊢ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c) ↔ ⟨{b}, ⟨{z}, x⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) |
14 | | elin 3220 |
. . . . . . . . . . . . 13
⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) ↔ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins2 Ins2 S ∧ ⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd )))) |
15 | | snex 4112 |
. . . . . . . . . . . . . . . 16
⊢ {b} ∈
V |
16 | 15 | otelins2 5792 |
. . . . . . . . . . . . . . 15
⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins2 Ins2 S ↔ ⟨{a}, ⟨{z}, x⟩⟩ ∈ Ins2 S ) |
17 | 5 | otelins2 5792 |
. . . . . . . . . . . . . . 15
⊢ (⟨{a}, ⟨{z}, x⟩⟩ ∈ Ins2 S ↔ ⟨{a}, x⟩ ∈ S
) |
18 | | vex 2863 |
. . . . . . . . . . . . . . . 16
⊢ a ∈
V |
19 | 18, 7 | opelssetsn 4761 |
. . . . . . . . . . . . . . 15
⊢ (⟨{a}, x⟩ ∈ S ↔ a ∈ x) |
20 | 16, 17, 19 | 3bitri 262 |
. . . . . . . . . . . . . 14
⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins2 Ins2 S ↔ a ∈ x) |
21 | 7 | oqelins4 5795 |
. . . . . . . . . . . . . . 15
⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd )) ↔ ⟨{a}, ⟨{b}, {z}⟩⟩ ∈ SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) |
22 | | vex 2863 |
. . . . . . . . . . . . . . . 16
⊢ z ∈
V |
23 | 18, 9, 22 | otsnelsi3 5806 |
. . . . . . . . . . . . . . 15
⊢ (⟨{a}, ⟨{b}, {z}⟩⟩ ∈ SI3 ( Ins2 ◡1st ∩ (V × ◡2nd )) ↔ ⟨a, ⟨b, z⟩⟩ ∈ ( Ins2 ◡1st ∩ (V × ◡2nd ))) |
24 | | elin 3220 |
. . . . . . . . . . . . . . . 16
⊢ (⟨a, ⟨b, z⟩⟩ ∈ ( Ins2 ◡1st ∩ (V × ◡2nd )) ↔ (⟨a, ⟨b, z⟩⟩ ∈ Ins2 ◡1st ∧ ⟨a, ⟨b, z⟩⟩ ∈ (V × ◡2nd ))) |
25 | 9 | otelins2 5792 |
. . . . . . . . . . . . . . . . . 18
⊢ (⟨a, ⟨b, z⟩⟩ ∈ Ins2 ◡1st ↔ ⟨a, z⟩ ∈ ◡1st ) |
26 | | df-br 4641 |
. . . . . . . . . . . . . . . . . 18
⊢ (a◡1st z ↔ ⟨a, z⟩ ∈ ◡1st ) |
27 | | brcnv 4893 |
. . . . . . . . . . . . . . . . . 18
⊢ (a◡1st z ↔ z1st a) |
28 | 25, 26, 27 | 3bitr2i 264 |
. . . . . . . . . . . . . . . . 17
⊢ (⟨a, ⟨b, z⟩⟩ ∈ Ins2 ◡1st ↔ z1st a) |
29 | | opelxp 4812 |
. . . . . . . . . . . . . . . . . . 19
⊢ (⟨a, ⟨b, z⟩⟩ ∈ (V ×
◡2nd ) ↔ (a ∈ V ∧ ⟨b, z⟩ ∈ ◡2nd )) |
30 | 18, 29 | mpbiran 884 |
. . . . . . . . . . . . . . . . . 18
⊢ (⟨a, ⟨b, z⟩⟩ ∈ (V ×
◡2nd ) ↔ ⟨b, z⟩ ∈ ◡2nd ) |
31 | | df-br 4641 |
. . . . . . . . . . . . . . . . . 18
⊢ (b◡2nd z ↔ ⟨b, z⟩ ∈ ◡2nd ) |
32 | | brcnv 4893 |
. . . . . . . . . . . . . . . . . 18
⊢ (b◡2nd z ↔ z2nd b) |
33 | 30, 31, 32 | 3bitr2i 264 |
. . . . . . . . . . . . . . . . 17
⊢ (⟨a, ⟨b, z⟩⟩ ∈ (V ×
◡2nd ) ↔ z2nd b) |
34 | 28, 33 | anbi12i 678 |
. . . . . . . . . . . . . . . 16
⊢ ((⟨a, ⟨b, z⟩⟩ ∈ Ins2 ◡1st ∧ ⟨a, ⟨b, z⟩⟩ ∈ (V × ◡2nd )) ↔ (z1st a ∧ z2nd b)) |
35 | 18, 9 | op1st2nd 5791 |
. . . . . . . . . . . . . . . 16
⊢ ((z1st a ∧ z2nd b) ↔ z =
⟨a,
b⟩) |
36 | 24, 34, 35 | 3bitri 262 |
. . . . . . . . . . . . . . 15
⊢ (⟨a, ⟨b, z⟩⟩ ∈ ( Ins2 ◡1st ∩ (V × ◡2nd )) ↔ z = ⟨a, b⟩) |
37 | 21, 23, 36 | 3bitri 262 |
. . . . . . . . . . . . . 14
⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd )) ↔ z = ⟨a, b⟩) |
38 | 20, 37 | anbi12i 678 |
. . . . . . . . . . . . 13
⊢ ((⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins2 Ins2 S ∧ ⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) ↔ (a ∈ x ∧ z = ⟨a, b⟩)) |
39 | 14, 38 | bitri 240 |
. . . . . . . . . . . 12
⊢ (⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) ↔ (a ∈ x ∧ z = ⟨a, b⟩)) |
40 | 39 | exbii 1582 |
. . . . . . . . . . 11
⊢ (∃a⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) ↔ ∃a(a ∈ x ∧ z = ⟨a, b⟩)) |
41 | | elima1c 4948 |
. . . . . . . . . . 11
⊢ (⟨{b}, ⟨{z}, x⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c) ↔ ∃a⟨{a}, ⟨{b}, ⟨{z}, x⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd )))) |
42 | | df-rex 2621 |
. . . . . . . . . . 11
⊢ (∃a ∈ x z = ⟨a, b⟩ ↔ ∃a(a ∈ x ∧ z = ⟨a, b⟩)) |
43 | 40, 41, 42 | 3bitr4i 268 |
. . . . . . . . . 10
⊢ (⟨{b}, ⟨{z}, x⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c) ↔ ∃a ∈ x z = ⟨a, b⟩) |
44 | 13, 43 | bitri 240 |
. . . . . . . . 9
⊢ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c) ↔ ∃a ∈ x z = ⟨a, b⟩) |
45 | 12, 44 | anbi12i 678 |
. . . . . . . 8
⊢ ((⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins2 Ins2 S ∧ ⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) ↔ (b ∈ y ∧ ∃a ∈ x z = ⟨a, b⟩)) |
46 | 4, 45 | bitri 240 |
. . . . . . 7
⊢ (⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) ↔ (b ∈ y ∧ ∃a ∈ x z = ⟨a, b⟩)) |
47 | 46 | exbii 1582 |
. . . . . 6
⊢ (∃b⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) ↔ ∃b(b ∈ y ∧ ∃a ∈ x z = ⟨a, b⟩)) |
48 | | elima1c 4948 |
. . . . . 6
⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) “ 1c) ↔ ∃b⟨{b}, ⟨{z}, ⟨x, y⟩⟩⟩ ∈ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c))) |
49 | | df-rex 2621 |
. . . . . 6
⊢ (∃b ∈ y ∃a ∈ x z = ⟨a, b⟩ ↔ ∃b(b ∈ y ∧ ∃a ∈ x z = ⟨a, b⟩)) |
50 | 47, 48, 49 | 3bitr4i 268 |
. . . . 5
⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) “ 1c) ↔ ∃b ∈ y ∃a ∈ x z = ⟨a, b⟩) |
51 | 2, 3, 50 | 3bitr4ri 269 |
. . . 4
⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) “ 1c) ↔ z ∈ (x × y)) |
52 | 51 | releqmpt2 5810 |
. . 3
⊢ (((V × V)
× V) ∖ (( Ins2 S ⊕ Ins3 (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) “ 1c)) “
1c)) = (x ∈ V, y ∈ V ↦ (x × y)) |
53 | 1, 52 | eqtr4i 2376 |
. 2
⊢ Cross = (((V × V) × V) ∖ (( Ins2 S ⊕ Ins3 (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) “ 1c)) “
1c)) |
54 | | vvex 4110 |
. . 3
⊢ V ∈ V |
55 | | ssetex 4745 |
. . . . . . 7
⊢ S ∈
V |
56 | 55 | ins2ex 5798 |
. . . . . 6
⊢ Ins2 S ∈ V |
57 | 56 | ins2ex 5798 |
. . . . 5
⊢ Ins2 Ins2 S ∈
V |
58 | | 1stex 4740 |
. . . . . . . . . . . . 13
⊢ 1st
∈ V |
59 | 58 | cnvex 5103 |
. . . . . . . . . . . 12
⊢ ◡1st ∈ V |
60 | 59 | ins2ex 5798 |
. . . . . . . . . . 11
⊢ Ins2 ◡1st ∈ V |
61 | | 2ndex 5113 |
. . . . . . . . . . . . 13
⊢ 2nd
∈ V |
62 | 61 | cnvex 5103 |
. . . . . . . . . . . 12
⊢ ◡2nd ∈ V |
63 | 54, 62 | xpex 5116 |
. . . . . . . . . . 11
⊢ (V × ◡2nd ) ∈ V |
64 | 60, 63 | inex 4106 |
. . . . . . . . . 10
⊢ ( Ins2 ◡1st ∩ (V × ◡2nd )) ∈ V |
65 | 64 | si3ex 5807 |
. . . . . . . . 9
⊢ SI3 ( Ins2 ◡1st ∩ (V × ◡2nd )) ∈ V |
66 | 65 | ins4ex 5800 |
. . . . . . . 8
⊢ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd )) ∈ V |
67 | 57, 66 | inex 4106 |
. . . . . . 7
⊢ ( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) ∈ V |
68 | | 1cex 4143 |
. . . . . . 7
⊢
1c ∈
V |
69 | 67, 68 | imaex 4748 |
. . . . . 6
⊢ (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c) ∈ V |
70 | 69 | ins4ex 5800 |
. . . . 5
⊢ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c) ∈ V |
71 | 57, 70 | inex 4106 |
. . . 4
⊢ ( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) ∈ V |
72 | 71, 68 | imaex 4748 |
. . 3
⊢ (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3 ( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) “ 1c) ∈ V |
73 | 54, 54, 72 | mpt2exlem 5812 |
. 2
⊢ (((V × V)
× V) ∖ (( Ins2 S ⊕ Ins3 (( Ins2 Ins2 S ∩ Ins4 (( Ins2 Ins2 S ∩ Ins4 SI3
( Ins2 ◡1st ∩ (V × ◡2nd ))) “
1c)) “ 1c)) “
1c)) ∈ V |
74 | 53, 73 | eqeltri 2423 |
1
⊢ Cross ∈
V |