Step | Hyp | Ref
| Expression |
1 | | ressbas.b |
. . . . 5
β’ π΅ = (Baseβπ) |
2 | | simp1 1137 |
. . . . . 6
β’ ((π΅ β π΄ β§ π β V β§ π΄ β π) β π΅ β π΄) |
3 | | sseqin2 4180 |
. . . . . 6
β’ (π΅ β π΄ β (π΄ β© π΅) = π΅) |
4 | 2, 3 | sylib 217 |
. . . . 5
β’ ((π΅ β π΄ β§ π β V β§ π΄ β π) β (π΄ β© π΅) = π΅) |
5 | | ressbas.r |
. . . . . . 7
β’ π
= (π βΎs π΄) |
6 | 5, 1 | ressid2 17123 |
. . . . . 6
β’ ((π΅ β π΄ β§ π β V β§ π΄ β π) β π
= π) |
7 | 6 | fveq2d 6851 |
. . . . 5
β’ ((π΅ β π΄ β§ π β V β§ π΄ β π) β (Baseβπ
) = (Baseβπ)) |
8 | 1, 4, 7 | 3eqtr4a 2803 |
. . . 4
β’ ((π΅ β π΄ β§ π β V β§ π΄ β π) β (π΄ β© π΅) = (Baseβπ
)) |
9 | 8 | 3expib 1123 |
. . 3
β’ (π΅ β π΄ β ((π β V β§ π΄ β π) β (π΄ β© π΅) = (Baseβπ
))) |
10 | | simp2 1138 |
. . . . . 6
β’ ((Β¬
π΅ β π΄ β§ π β V β§ π΄ β π) β π β V) |
11 | 1 | fvexi 6861 |
. . . . . . 7
β’ π΅ β V |
12 | 11 | inex2 5280 |
. . . . . 6
β’ (π΄ β© π΅) β V |
13 | | baseid 17093 |
. . . . . . 7
β’ Base =
Slot (Baseβndx) |
14 | 13 | setsid 17087 |
. . . . . 6
β’ ((π β V β§ (π΄ β© π΅) β V) β (π΄ β© π΅) = (Baseβ(π sSet β¨(Baseβndx), (π΄ β© π΅)β©))) |
15 | 10, 12, 14 | sylancl 587 |
. . . . 5
β’ ((Β¬
π΅ β π΄ β§ π β V β§ π΄ β π) β (π΄ β© π΅) = (Baseβ(π sSet β¨(Baseβndx), (π΄ β© π΅)β©))) |
16 | 5, 1 | ressval2 17124 |
. . . . . 6
β’ ((Β¬
π΅ β π΄ β§ π β V β§ π΄ β π) β π
= (π sSet β¨(Baseβndx), (π΄ β© π΅)β©)) |
17 | 16 | fveq2d 6851 |
. . . . 5
β’ ((Β¬
π΅ β π΄ β§ π β V β§ π΄ β π) β (Baseβπ
) = (Baseβ(π sSet β¨(Baseβndx), (π΄ β© π΅)β©))) |
18 | 15, 17 | eqtr4d 2780 |
. . . 4
β’ ((Β¬
π΅ β π΄ β§ π β V β§ π΄ β π) β (π΄ β© π΅) = (Baseβπ
)) |
19 | 18 | 3expib 1123 |
. . 3
β’ (Β¬
π΅ β π΄ β ((π β V β§ π΄ β π) β (π΄ β© π΅) = (Baseβπ
))) |
20 | 9, 19 | pm2.61i 182 |
. 2
β’ ((π β V β§ π΄ β π) β (π΄ β© π΅) = (Baseβπ
)) |
21 | | in0 4356 |
. . . . 5
β’ (π΄ β© β
) =
β
|
22 | | fvprc 6839 |
. . . . . . 7
β’ (Β¬
π β V β
(Baseβπ) =
β
) |
23 | 1, 22 | eqtrid 2789 |
. . . . . 6
β’ (Β¬
π β V β π΅ = β
) |
24 | 23 | ineq2d 4177 |
. . . . 5
β’ (Β¬
π β V β (π΄ β© π΅) = (π΄ β© β
)) |
25 | 21, 24, 22 | 3eqtr4a 2803 |
. . . 4
β’ (Β¬
π β V β (π΄ β© π΅) = (Baseβπ)) |
26 | | base0 17095 |
. . . . . 6
β’ β
=
(Baseββ
) |
27 | 26 | eqcomi 2746 |
. . . . 5
β’
(Baseββ
) = β
|
28 | | reldmress 17121 |
. . . . 5
β’ Rel dom
βΎs |
29 | 27, 5, 28 | oveqprc 17071 |
. . . 4
β’ (Β¬
π β V β
(Baseβπ) =
(Baseβπ
)) |
30 | 25, 29 | eqtrd 2777 |
. . 3
β’ (Β¬
π β V β (π΄ β© π΅) = (Baseβπ
)) |
31 | 30 | adantr 482 |
. 2
β’ ((Β¬
π β V β§ π΄ β π) β (π΄ β© π΅) = (Baseβπ
)) |
32 | 20, 31 | pm2.61ian 811 |
1
β’ (π΄ β π β (π΄ β© π΅) = (Baseβπ
)) |