Step | Hyp | Ref
| Expression |
1 | | updjud.a |
. . . . . 6
β’ (π β π΄ β π) |
2 | | updjud.b |
. . . . . 6
β’ (π β π΅ β π) |
3 | 1, 2 | jca 512 |
. . . . 5
β’ (π β (π΄ β π β§ π΅ β π)) |
4 | | djuex 9844 |
. . . . 5
β’ ((π΄ β π β§ π΅ β π) β (π΄ β π΅) β V) |
5 | | mptexg 7171 |
. . . . 5
β’ ((π΄ β π΅) β V β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β V) |
6 | 3, 4, 5 | 3syl 18 |
. . . 4
β’ (π β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β V) |
7 | | feq1 6649 |
. . . . . . 7
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (β:(π΄ β π΅)βΆπΆ β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ)) |
8 | | coeq1 5813 |
. . . . . . . 8
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (β β (inl βΎ π΄)) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄))) |
9 | 8 | eqeq1d 2738 |
. . . . . . 7
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β ((β β (inl βΎ π΄)) = πΉ β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ)) |
10 | | coeq1 5813 |
. . . . . . . 8
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (β β (inr βΎ π΅)) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅))) |
11 | 10 | eqeq1d 2738 |
. . . . . . 7
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β ((β β (inr βΎ π΅)) = πΊ β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) |
12 | 7, 9, 11 | 3anbi123d 1436 |
. . . . . 6
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β ((β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ))) |
13 | | eqeq1 2740 |
. . . . . . . 8
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (β = π β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π)) |
14 | 13 | imbi2d 340 |
. . . . . . 7
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β β = π) β ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π))) |
15 | 14 | ralbidv 3174 |
. . . . . 6
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β β = π) β βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π))) |
16 | 12, 15 | anbi12d 631 |
. . . . 5
β’ (β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (((β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ) β§ βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β β = π)) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) β§ βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π)))) |
17 | 16 | adantl 482 |
. . . 4
β’ ((π β§ β = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))) β (((β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ) β§ βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β β = π)) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) β§ βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π)))) |
18 | | updjud.f |
. . . . . 6
β’ (π β πΉ:π΄βΆπΆ) |
19 | | updjud.g |
. . . . . 6
β’ (π β πΊ:π΅βΆπΆ) |
20 | | eqid 2736 |
. . . . . 6
β’ (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) |
21 | 18, 19, 20 | updjudhf 9867 |
. . . . 5
β’ (π β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ) |
22 | 18, 19, 20 | updjudhcoinlf 9868 |
. . . . 5
β’ (π β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ) |
23 | 18, 19, 20 | updjudhcoinrg 9869 |
. . . . 5
β’ (π β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) |
24 | | simpr 485 |
. . . . . . 7
β’ ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) |
25 | | eqeq2 2748 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β ((π β (inl βΎ π΄)) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) β (π β (inl βΎ π΄)) = πΉ)) |
26 | | fvres 6861 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (π§ β π΄ β ((inl βΎ π΄)βπ§) = (inlβπ§)) |
27 | 26 | eqcomd 2742 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π§ β π΄ β (inlβπ§) = ((inl βΎ π΄)βπ§)) |
28 | 27 | eqeq2d 2747 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π§ β π΄ β (π¦ = (inlβπ§) β π¦ = ((inl βΎ π΄)βπ§))) |
29 | 28 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β§ π) β§ π§ β π΄) β (π¦ = (inlβπ§) β π¦ = ((inl βΎ π΄)βπ§))) |
30 | | fveq1 6841 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄))βπ§) = ((π β (inl βΎ π΄))βπ§)) |
31 | 30 | ad2antrr 724 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β§ π) β§ π§ β π΄) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄))βπ§) = ((π β (inl βΎ π΄))βπ§)) |
32 | | inlresf 9850 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ (inl
βΎ π΄):π΄βΆ(π΄ β π΅) |
33 | | ffn 6668 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((inl
βΎ π΄):π΄βΆ(π΄ β π΅) β (inl βΎ π΄) Fn π΄) |
34 | 32, 33 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ ((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β§ π) β (inl βΎ π΄) Fn π΄) |
35 | | fvco2 6938 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (((inl
βΎ π΄) Fn π΄ β§ π§ β π΄) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄))βπ§) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inl βΎ π΄)βπ§))) |
36 | 34, 35 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β§ π) β§ π§ β π΄) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄))βπ§) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inl βΎ π΄)βπ§))) |
37 | | fvco2 6938 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (((inl
βΎ π΄) Fn π΄ β§ π§ β π΄) β ((π β (inl βΎ π΄))βπ§) = (πβ((inl βΎ π΄)βπ§))) |
38 | 34, 37 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β§ π) β§ π§ β π΄) β ((π β (inl βΎ π΄))βπ§) = (πβ((inl βΎ π΄)βπ§))) |
39 | 31, 36, 38 | 3eqtr3d 2784 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β§ π) β§ π§ β π΄) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inl βΎ π΄)βπ§)) = (πβ((inl βΎ π΄)βπ§))) |
40 | | fveq2 6842 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π¦ = ((inl βΎ π΄)βπ§) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inl βΎ π΄)βπ§))) |
41 | | fveq2 6842 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π¦ = ((inl βΎ π΄)βπ§) β (πβπ¦) = (πβ((inl βΎ π΄)βπ§))) |
42 | 40, 41 | eqeq12d 2752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π¦ = ((inl βΎ π΄)βπ§) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inl βΎ π΄)βπ§)) = (πβ((inl βΎ π΄)βπ§)))) |
43 | 39, 42 | syl5ibrcom 246 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β§ π) β§ π§ β π΄) β (π¦ = ((inl βΎ π΄)βπ§) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
44 | 29, 43 | sylbid 239 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β§ π) β§ π§ β π΄) β (π¦ = (inlβπ§) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
45 | 44 | expimpd 454 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β§ π) β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
46 | 45 | ex 413 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = (π β (inl βΎ π΄)) β (π β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
47 | 46 | eqcoms 2744 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π β (inl βΎ π΄)) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) β (π β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
48 | 25, 47 | syl6bir 253 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β ((π β (inl βΎ π΄)) = πΉ β (π β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))))) |
49 | 48 | com23 86 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β (π β ((π β (inl βΎ π΄)) = πΉ β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))))) |
50 | 49 | 3ad2ant2 1134 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) β (π β ((π β (inl βΎ π΄)) = πΉ β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))))) |
51 | 50 | impcom 408 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β ((π β (inl βΎ π΄)) = πΉ β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
52 | 51 | com12 32 |
. . . . . . . . . . . . . . . . 17
β’ ((π β (inl βΎ π΄)) = πΉ β ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
53 | 52 | 3ad2ant2 1134 |
. . . . . . . . . . . . . . . 16
β’ ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
54 | 53 | impcom 408 |
. . . . . . . . . . . . . . 15
β’ (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β ((π§ β π΄ β§ π¦ = (inlβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
55 | 54 | com12 32 |
. . . . . . . . . . . . . 14
β’ ((π§ β π΄ β§ π¦ = (inlβπ§)) β (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
56 | 55 | rexlimiva 3144 |
. . . . . . . . . . . . 13
β’
(βπ§ β
π΄ π¦ = (inlβπ§) β (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
57 | | eqeq2 2748 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ β ((π β (inr βΎ π΅)) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) β (π β (inr βΎ π΅)) = πΊ)) |
58 | | fvres 6861 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (π§ β π΅ β ((inr βΎ π΅)βπ§) = (inrβπ§)) |
59 | 58 | eqcomd 2742 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π§ β π΅ β (inrβπ§) = ((inr βΎ π΅)βπ§)) |
60 | 59 | eqeq2d 2747 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π§ β π΅ β (π¦ = (inrβπ§) β π¦ = ((inr βΎ π΅)βπ§))) |
61 | 60 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β§ π) β§ π§ β π΅) β (π¦ = (inrβπ§) β π¦ = ((inr βΎ π΅)βπ§))) |
62 | | fveq1 6841 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅))βπ§) = ((π β (inr βΎ π΅))βπ§)) |
63 | 62 | ad2antrr 724 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β§ π) β§ π§ β π΅) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅))βπ§) = ((π β (inr βΎ π΅))βπ§)) |
64 | | inrresf 9852 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ (inr
βΎ π΅):π΅βΆ(π΄ β π΅) |
65 | | ffn 6668 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((inr
βΎ π΅):π΅βΆ(π΄ β π΅) β (inr βΎ π΅) Fn π΅) |
66 | 64, 65 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ ((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β§ π) β (inr βΎ π΅) Fn π΅) |
67 | | fvco2 6938 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (((inr
βΎ π΅) Fn π΅ β§ π§ β π΅) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅))βπ§) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inr βΎ π΅)βπ§))) |
68 | 66, 67 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β§ π) β§ π§ β π΅) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅))βπ§) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inr βΎ π΅)βπ§))) |
69 | | fvco2 6938 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (((inr
βΎ π΅) Fn π΅ β§ π§ β π΅) β ((π β (inr βΎ π΅))βπ§) = (πβ((inr βΎ π΅)βπ§))) |
70 | 66, 69 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β§ π) β§ π§ β π΅) β ((π β (inr βΎ π΅))βπ§) = (πβ((inr βΎ π΅)βπ§))) |
71 | 63, 68, 70 | 3eqtr3d 2784 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β§ π) β§ π§ β π΅) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inr βΎ π΅)βπ§)) = (πβ((inr βΎ π΅)βπ§))) |
72 | | fveq2 6842 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π¦ = ((inr βΎ π΅)βπ§) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inr βΎ π΅)βπ§))) |
73 | | fveq2 6842 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π¦ = ((inr βΎ π΅)βπ§) β (πβπ¦) = (πβ((inr βΎ π΅)βπ§))) |
74 | 72, 73 | eqeq12d 2752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π¦ = ((inr βΎ π΅)βπ§) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))β((inr βΎ π΅)βπ§)) = (πβ((inr βΎ π΅)βπ§)))) |
75 | 71, 74 | syl5ibrcom 246 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β§ π) β§ π§ β π΅) β (π¦ = ((inr βΎ π΅)βπ§) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
76 | 61, 75 | sylbid 239 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’
(((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β§ π) β§ π§ β π΅) β (π¦ = (inrβπ§) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
77 | 76 | expimpd 454 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β§ π) β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
78 | 77 | ex 413 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = (π β (inr βΎ π΅)) β (π β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
79 | 78 | eqcoms 2744 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π β (inr βΎ π΅)) = ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) β (π β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
80 | 57, 79 | syl6bir 253 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ β ((π β (inr βΎ π΅)) = πΊ β (π β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))))) |
81 | 80 | com23 86 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ β (π β ((π β (inr βΎ π΅)) = πΊ β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))))) |
82 | 81 | 3ad2ant3 1135 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) β (π β ((π β (inr βΎ π΅)) = πΊ β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))))) |
83 | 82 | impcom 408 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β ((π β (inr βΎ π΅)) = πΊ β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
84 | 83 | com12 32 |
. . . . . . . . . . . . . . . . 17
β’ ((π β (inr βΎ π΅)) = πΊ β ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
85 | 84 | 3ad2ant3 1135 |
. . . . . . . . . . . . . . . 16
β’ ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)))) |
86 | 85 | impcom 408 |
. . . . . . . . . . . . . . 15
β’ (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β ((π§ β π΅ β§ π¦ = (inrβπ§)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
87 | 86 | com12 32 |
. . . . . . . . . . . . . 14
β’ ((π§ β π΅ β§ π¦ = (inrβπ§)) β (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
88 | 87 | rexlimiva 3144 |
. . . . . . . . . . . . 13
β’
(βπ§ β
π΅ π¦ = (inrβπ§) β (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
89 | 56, 88 | jaoi 855 |
. . . . . . . . . . . 12
β’
((βπ§ β
π΄ π¦ = (inlβπ§) β¨ βπ§ β π΅ π¦ = (inrβπ§)) β (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
90 | | djur 9855 |
. . . . . . . . . . . 12
β’ (π¦ β (π΄ β π΅) β (βπ§ β π΄ π¦ = (inlβπ§) β¨ βπ§ β π΅ π¦ = (inrβπ§))) |
91 | 89, 90 | syl11 33 |
. . . . . . . . . . 11
β’ (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β (π¦ β (π΄ β π΅) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
92 | 91 | ralrimiv 3142 |
. . . . . . . . . 10
β’ (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β βπ¦ β (π΄ β π΅)((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦)) |
93 | | ffn 6668 |
. . . . . . . . . . . . 13
β’ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) Fn (π΄ β π΅)) |
94 | 93 | 3ad2ant1 1133 |
. . . . . . . . . . . 12
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) Fn (π΄ β π΅)) |
95 | 94 | adantl 482 |
. . . . . . . . . . 11
β’ ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) Fn (π΄ β π΅)) |
96 | | ffn 6668 |
. . . . . . . . . . . 12
β’ (π:(π΄ β π΅)βΆπΆ β π Fn (π΄ β π΅)) |
97 | 96 | 3ad2ant1 1133 |
. . . . . . . . . . 11
β’ ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β π Fn (π΄ β π΅)) |
98 | | eqfnfv 6982 |
. . . . . . . . . . 11
β’ (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) Fn (π΄ β π΅) β§ π Fn (π΄ β π΅)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π β βπ¦ β (π΄ β π΅)((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
99 | 95, 97, 98 | syl2an 596 |
. . . . . . . . . 10
β’ (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π β βπ¦ β (π΄ β π΅)((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯))))βπ¦) = (πβπ¦))) |
100 | 92, 99 | mpbird 256 |
. . . . . . . . 9
β’ (((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β§ (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ)) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π) |
101 | 100 | ex 413 |
. . . . . . . 8
β’ ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π)) |
102 | 101 | ralrimivw 3147 |
. . . . . . 7
β’ ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π)) |
103 | 24, 102 | jca 512 |
. . . . . 6
β’ ((π β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ)) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) β§ βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π))) |
104 | 103 | ex 413 |
. . . . 5
β’ (π β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) β§ βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π)))) |
105 | 21, 22, 23, 104 | mp3and 1464 |
. . . 4
β’ (π β (((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))):(π΄ β π΅)βΆπΆ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inl βΎ π΄)) = πΉ β§ ((π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β (inr βΎ π΅)) = πΊ) β§ βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β
, (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) = π))) |
106 | 6, 17, 105 | rspcedvd 3583 |
. . 3
β’ (π β ββ β V ((β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ) β§ βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β β = π))) |
107 | | feq1 6649 |
. . . . 5
β’ (β = π β (β:(π΄ β π΅)βΆπΆ β π:(π΄ β π΅)βΆπΆ)) |
108 | | coeq1 5813 |
. . . . . 6
β’ (β = π β (β β (inl βΎ π΄)) = (π β (inl βΎ π΄))) |
109 | 108 | eqeq1d 2738 |
. . . . 5
β’ (β = π β ((β β (inl βΎ π΄)) = πΉ β (π β (inl βΎ π΄)) = πΉ)) |
110 | | coeq1 5813 |
. . . . . 6
β’ (β = π β (β β (inr βΎ π΅)) = (π β (inr βΎ π΅))) |
111 | 110 | eqeq1d 2738 |
. . . . 5
β’ (β = π β ((β β (inr βΎ π΅)) = πΊ β (π β (inr βΎ π΅)) = πΊ)) |
112 | 107, 109,
111 | 3anbi123d 1436 |
. . . 4
β’ (β = π β ((β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ) β (π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ))) |
113 | 112 | reu8 3691 |
. . 3
β’
(β!β β V
(β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ) β ββ β V ((β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ) β§ βπ β V ((π:(π΄ β π΅)βΆπΆ β§ (π β (inl βΎ π΄)) = πΉ β§ (π β (inr βΎ π΅)) = πΊ) β β = π))) |
114 | 106, 113 | sylibr 233 |
. 2
β’ (π β β!β β V (β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ)) |
115 | | reuv 3471 |
. 2
β’
(β!β β V
(β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ) β β!β(β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ)) |
116 | 114, 115 | sylib 217 |
1
β’ (π β β!β(β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ)) |