Metamath Proof Explorer


Theorem fucsect

Description: Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017)

Ref Expression
Hypotheses fuciso.q Q = C FuncCat D
fuciso.b B = Base C
fuciso.n N = C Nat D
fuciso.f φ F C Func D
fuciso.g φ G C Func D
fucsect.s S = Sect Q
fucsect.t T = Sect D
Assertion fucsect φ U F S G V U F N G V G N F x B U x 1 st F x T 1 st G x V x

Proof

Step Hyp Ref Expression
1 fuciso.q Q = C FuncCat D
2 fuciso.b B = Base C
3 fuciso.n N = C Nat D
4 fuciso.f φ F C Func D
5 fuciso.g φ G C Func D
6 fucsect.s S = Sect Q
7 fucsect.t T = Sect D
8 1 fucbas C Func D = Base Q
9 1 3 fuchom N = Hom Q
10 eqid comp Q = comp Q
11 eqid Id Q = Id Q
12 funcrcl F C Func D C Cat D Cat
13 4 12 syl φ C Cat D Cat
14 13 simpld φ C Cat
15 13 simprd φ D Cat
16 1 14 15 fuccat φ Q Cat
17 8 9 10 11 6 16 4 5 issect φ U F S G V U F N G V G N F V F G comp Q F U = Id Q F
18 ovex V x 1 st F x 1 st G x comp D 1 st F x U x V
19 18 rgenw x B V x 1 st F x 1 st G x comp D 1 st F x U x V
20 mpteqb x B V x 1 st F x 1 st G x comp D 1 st F x U x V x B V x 1 st F x 1 st G x comp D 1 st F x U x = x B Id D 1 st F x x B V x 1 st F x 1 st G x comp D 1 st F x U x = Id D 1 st F x
21 19 20 mp1i φ U F N G V G N F x B V x 1 st F x 1 st G x comp D 1 st F x U x = x B Id D 1 st F x x B V x 1 st F x 1 st G x comp D 1 st F x U x = Id D 1 st F x
22 eqid comp D = comp D
23 simprl φ U F N G V G N F U F N G
24 simprr φ U F N G V G N F V G N F
25 1 3 2 22 10 23 24 fucco φ U F N G V G N F V F G comp Q F U = x B V x 1 st F x 1 st G x comp D 1 st F x U x
26 eqid Id D = Id D
27 4 adantr φ U F N G V G N F F C Func D
28 1 11 26 27 fucid φ U F N G V G N F Id Q F = Id D 1 st F
29 15 adantr φ U F N G V G N F D Cat
30 eqid Base D = Base D
31 30 26 cidfn D Cat Id D Fn Base D
32 29 31 syl φ U F N G V G N F Id D Fn Base D
33 dffn2 Id D Fn Base D Id D : Base D V
34 32 33 sylib φ U F N G V G N F Id D : Base D V
35 relfunc Rel C Func D
36 1st2ndbr Rel C Func D F C Func D 1 st F C Func D 2 nd F
37 35 4 36 sylancr φ 1 st F C Func D 2 nd F
38 2 30 37 funcf1 φ 1 st F : B Base D
39 38 adantr φ U F N G V G N F 1 st F : B Base D
40 fcompt Id D : Base D V 1 st F : B Base D Id D 1 st F = x B Id D 1 st F x
41 34 39 40 syl2anc φ U F N G V G N F Id D 1 st F = x B Id D 1 st F x
42 28 41 eqtrd φ U F N G V G N F Id Q F = x B Id D 1 st F x
43 25 42 eqeq12d φ U F N G V G N F V F G comp Q F U = Id Q F x B V x 1 st F x 1 st G x comp D 1 st F x U x = x B Id D 1 st F x
44 eqid Hom D = Hom D
45 29 adantr φ U F N G V G N F x B D Cat
46 39 ffvelrnda φ U F N G V G N F x B 1 st F x Base D
47 1st2ndbr Rel C Func D G C Func D 1 st G C Func D 2 nd G
48 35 5 47 sylancr φ 1 st G C Func D 2 nd G
49 2 30 48 funcf1 φ 1 st G : B Base D
50 49 adantr φ U F N G V G N F 1 st G : B Base D
51 50 ffvelrnda φ U F N G V G N F x B 1 st G x Base D
52 23 adantr φ U F N G V G N F x B U F N G
53 3 52 nat1st2nd φ U F N G V G N F x B U 1 st F 2 nd F N 1 st G 2 nd G
54 simpr φ U F N G V G N F x B x B
55 3 53 2 44 54 natcl φ U F N G V G N F x B U x 1 st F x Hom D 1 st G x
56 24 adantr φ U F N G V G N F x B V G N F
57 3 56 nat1st2nd φ U F N G V G N F x B V 1 st G 2 nd G N 1 st F 2 nd F
58 3 57 2 44 54 natcl φ U F N G V G N F x B V x 1 st G x Hom D 1 st F x
59 30 44 22 26 7 45 46 51 55 58 issect2 φ U F N G V G N F x B U x 1 st F x T 1 st G x V x V x 1 st F x 1 st G x comp D 1 st F x U x = Id D 1 st F x
60 59 ralbidva φ U F N G V G N F x B U x 1 st F x T 1 st G x V x x B V x 1 st F x 1 st G x comp D 1 st F x U x = Id D 1 st F x
61 21 43 60 3bitr4d φ U F N G V G N F V F G comp Q F U = Id Q F x B U x 1 st F x T 1 st G x V x
62 61 pm5.32da φ U F N G V G N F V F G comp Q F U = Id Q F U F N G V G N F x B U x 1 st F x T 1 st G x V x
63 df-3an U F N G V G N F V F G comp Q F U = Id Q F U F N G V G N F V F G comp Q F U = Id Q F
64 df-3an U F N G V G N F x B U x 1 st F x T 1 st G x V x U F N G V G N F x B U x 1 st F x T 1 st G x V x
65 62 63 64 3bitr4g φ U F N G V G N F V F G comp Q F U = Id Q F U F N G V G N F x B U x 1 st F x T 1 st G x V x
66 17 65 bitrd φ U F S G V U F N G V G N F x B U x 1 st F x T 1 st G x V x