Description: Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | prfval.k | |
|
prfval.b | |
||
prfval.h | |
||
prfval.c | |
||
prfval.d | |
||
Assertion | prfval | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prfval.k | |
|
2 | prfval.b | |
|
3 | prfval.h | |
|
4 | prfval.c | |
|
5 | prfval.d | |
|
6 | df-prf | |
|
7 | 6 | a1i | |
8 | fvex | |
|
9 | 8 | dmex | |
10 | 9 | a1i | |
11 | simprl | |
|
12 | 11 | fveq2d | |
13 | 12 | dmeqd | |
14 | eqid | |
|
15 | relfunc | |
|
16 | 1st2ndbr | |
|
17 | 15 4 16 | sylancr | |
18 | 2 14 17 | funcf1 | |
19 | 18 | fdmd | |
20 | 19 | adantr | |
21 | 13 20 | eqtrd | |
22 | simpr | |
|
23 | simplrl | |
|
24 | 23 | fveq2d | |
25 | 24 | fveq1d | |
26 | simplrr | |
|
27 | 26 | fveq2d | |
28 | 27 | fveq1d | |
29 | 25 28 | opeq12d | |
30 | 22 29 | mpteq12dv | |
31 | eqidd | |
|
32 | 22 22 31 | mpoeq123dv | |
33 | 23 | ad2antrr | |
34 | 33 | fveq2d | |
35 | 34 | oveqd | |
36 | 35 | dmeqd | |
37 | eqid | |
|
38 | 17 | ad4antr | |
39 | simplr | |
|
40 | simpr | |
|
41 | 2 3 37 38 39 40 | funcf2 | |
42 | 41 | fdmd | |
43 | 36 42 | eqtrd | |
44 | 35 | fveq1d | |
45 | 26 | ad2antrr | |
46 | 45 | fveq2d | |
47 | 46 | oveqd | |
48 | 47 | fveq1d | |
49 | 44 48 | opeq12d | |
50 | 43 49 | mpteq12dv | |
51 | 50 | 3impa | |
52 | 51 | mpoeq3dva | |
53 | 32 52 | eqtrd | |
54 | 30 53 | opeq12d | |
55 | 10 21 54 | csbied2 | |
56 | 4 | elexd | |
57 | 5 | elexd | |
58 | opex | |
|
59 | 58 | a1i | |
60 | 7 55 56 57 59 | ovmpod | |
61 | 1 60 | eqtrid | |