Step |
Hyp |
Ref |
Expression |
1 |
|
yoneda.y |
⊢ 𝑌 = ( Yon ‘ 𝐶 ) |
2 |
|
yoneda.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
yoneda.1 |
⊢ 1 = ( Id ‘ 𝐶 ) |
4 |
|
yoneda.o |
⊢ 𝑂 = ( oppCat ‘ 𝐶 ) |
5 |
|
yoneda.s |
⊢ 𝑆 = ( SetCat ‘ 𝑈 ) |
6 |
|
yoneda.t |
⊢ 𝑇 = ( SetCat ‘ 𝑉 ) |
7 |
|
yoneda.q |
⊢ 𝑄 = ( 𝑂 FuncCat 𝑆 ) |
8 |
|
yoneda.h |
⊢ 𝐻 = ( HomF ‘ 𝑄 ) |
9 |
|
yoneda.r |
⊢ 𝑅 = ( ( 𝑄 ×c 𝑂 ) FuncCat 𝑇 ) |
10 |
|
yoneda.e |
⊢ 𝐸 = ( 𝑂 evalF 𝑆 ) |
11 |
|
yoneda.z |
⊢ 𝑍 = ( 𝐻 ∘func ( ( 〈 ( 1st ‘ 𝑌 ) , tpos ( 2nd ‘ 𝑌 ) 〉 ∘func ( 𝑄 2ndF 𝑂 ) ) 〈,〉F ( 𝑄 1stF 𝑂 ) ) ) |
12 |
|
yoneda.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
13 |
|
yoneda.w |
⊢ ( 𝜑 → 𝑉 ∈ 𝑊 ) |
14 |
|
yoneda.u |
⊢ ( 𝜑 → ran ( Homf ‘ 𝐶 ) ⊆ 𝑈 ) |
15 |
|
yoneda.v |
⊢ ( 𝜑 → ( ran ( Homf ‘ 𝑄 ) ∪ 𝑈 ) ⊆ 𝑉 ) |
16 |
|
yonedalem21.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑂 Func 𝑆 ) ) |
17 |
|
yonedalem21.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
18 |
|
yonedalem4.n |
⊢ 𝑁 = ( 𝑓 ∈ ( 𝑂 Func 𝑆 ) , 𝑥 ∈ 𝐵 ↦ ( 𝑢 ∈ ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) ↦ ( 𝑦 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ↦ ( ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ 𝑔 ) ‘ 𝑢 ) ) ) ) ) |
19 |
|
yonedalem4.p |
⊢ ( 𝜑 → 𝐴 ∈ ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ) |
20 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
|
yonedalem4a |
⊢ ( 𝜑 → ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) = ( 𝑦 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) ) |
21 |
|
oveq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑋 ) = ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
22 |
|
oveq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑦 ) = ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ) |
23 |
22
|
fveq1d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑔 ) = ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ) |
24 |
23
|
fveq1d |
⊢ ( 𝑦 = 𝑧 → ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑔 ) ‘ 𝐴 ) = ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) |
25 |
21 24
|
mpteq12dv |
⊢ ( 𝑦 = 𝑧 → ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑔 ) ‘ 𝐴 ) ) = ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) |
26 |
25
|
cbvmptv |
⊢ ( 𝑦 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) = ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) |
27 |
20 26
|
eqtrdi |
⊢ ( 𝜑 → ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) = ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) ) |
28 |
4 2
|
oppcbas |
⊢ 𝐵 = ( Base ‘ 𝑂 ) |
29 |
|
eqid |
⊢ ( Hom ‘ 𝑂 ) = ( Hom ‘ 𝑂 ) |
30 |
|
eqid |
⊢ ( Hom ‘ 𝑆 ) = ( Hom ‘ 𝑆 ) |
31 |
|
relfunc |
⊢ Rel ( 𝑂 Func 𝑆 ) |
32 |
|
1st2ndbr |
⊢ ( ( Rel ( 𝑂 Func 𝑆 ) ∧ 𝐹 ∈ ( 𝑂 Func 𝑆 ) ) → ( 1st ‘ 𝐹 ) ( 𝑂 Func 𝑆 ) ( 2nd ‘ 𝐹 ) ) |
33 |
31 16 32
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) ( 𝑂 Func 𝑆 ) ( 2nd ‘ 𝐹 ) ) |
34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( 1st ‘ 𝐹 ) ( 𝑂 Func 𝑆 ) ( 2nd ‘ 𝐹 ) ) |
35 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
36 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → 𝑧 ∈ 𝐵 ) |
37 |
28 29 30 34 35 36
|
funcf2 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) : ( 𝑋 ( Hom ‘ 𝑂 ) 𝑧 ) ⟶ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
38 |
37
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) : ( 𝑋 ( Hom ‘ 𝑂 ) 𝑧 ) ⟶ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
39 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
40 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
41 |
40 4
|
oppchom |
⊢ ( 𝑋 ( Hom ‘ 𝑂 ) 𝑧 ) = ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) |
42 |
39 41
|
eleqtrrdi |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑔 ∈ ( 𝑋 ( Hom ‘ 𝑂 ) 𝑧 ) ) |
43 |
38 42
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ∈ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
44 |
15
|
unssbd |
⊢ ( 𝜑 → 𝑈 ⊆ 𝑉 ) |
45 |
13 44
|
ssexd |
⊢ ( 𝜑 → 𝑈 ∈ V ) |
46 |
45
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → 𝑈 ∈ V ) |
47 |
46
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑈 ∈ V ) |
48 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
49 |
28 48 33
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) |
50 |
5 45
|
setcbas |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝑆 ) ) |
51 |
50
|
feq3d |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐹 ) : 𝐵 ⟶ 𝑈 ↔ ( 1st ‘ 𝐹 ) : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) ) |
52 |
49 51
|
mpbird |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) : 𝐵 ⟶ 𝑈 ) |
53 |
52 17
|
ffvelrnd |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ∈ 𝑈 ) |
54 |
53
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ∈ 𝑈 ) |
55 |
52
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ∈ 𝑈 ) |
56 |
55
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ∈ 𝑈 ) |
57 |
5 47 30 54 56
|
elsetchom |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ∈ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ↔ ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) : ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
58 |
43 57
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) : ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) |
59 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝐴 ∈ ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ) |
60 |
58 59
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ∈ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) |
61 |
60
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) : ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) |
62 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → 𝐶 ∈ Cat ) |
63 |
1 2 62 35 40 36
|
yon11 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) = ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
64 |
63
|
feq2d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ↔ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) : ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
65 |
61 64
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) |
66 |
1 2 12 17 4 5 45 14
|
yon1cl |
⊢ ( 𝜑 → ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ∈ ( 𝑂 Func 𝑆 ) ) |
67 |
|
1st2ndbr |
⊢ ( ( Rel ( 𝑂 Func 𝑆 ) ∧ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ∈ ( 𝑂 Func 𝑆 ) ) → ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ( 𝑂 Func 𝑆 ) ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ) |
68 |
31 66 67
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ( 𝑂 Func 𝑆 ) ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ) |
69 |
28 48 68
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) |
70 |
50
|
feq3d |
⊢ ( 𝜑 → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) : 𝐵 ⟶ 𝑈 ↔ ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) ) |
71 |
69 70
|
mpbird |
⊢ ( 𝜑 → ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) : 𝐵 ⟶ 𝑈 ) |
72 |
71
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ∈ 𝑈 ) |
73 |
5 46 30 72 55
|
elsetchom |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ∈ ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ↔ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
74 |
65 73
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ∈ ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
75 |
74
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ∈ ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
76 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
77 |
|
mptelixpg |
⊢ ( 𝐵 ∈ V → ( ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) ∈ X 𝑧 ∈ 𝐵 ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝐵 ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ∈ ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) ) |
78 |
76 77
|
ax-mp |
⊢ ( ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) ∈ X 𝑧 ∈ 𝐵 ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ 𝐵 ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ∈ ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
79 |
75 78
|
sylibr |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) ∈ X 𝑧 ∈ 𝐵 ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
80 |
27 79
|
eqeltrd |
⊢ ( 𝜑 → ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ∈ X 𝑧 ∈ 𝐵 ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
81 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → 𝐶 ∈ Cat ) |
82 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → 𝑋 ∈ 𝐵 ) |
83 |
|
simpr1 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → 𝑧 ∈ 𝐵 ) |
84 |
1 2 81 82 40 83
|
yon11 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) = ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
85 |
84
|
eleq2d |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( 𝑘 ∈ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ↔ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) ) |
86 |
85
|
biimpa |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ) → 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
87 |
|
eqid |
⊢ ( comp ‘ 𝑂 ) = ( comp ‘ 𝑂 ) |
88 |
|
eqid |
⊢ ( comp ‘ 𝑆 ) = ( comp ‘ 𝑆 ) |
89 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( 1st ‘ 𝐹 ) ( 𝑂 Func 𝑆 ) ( 2nd ‘ 𝐹 ) ) |
90 |
89
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( 1st ‘ 𝐹 ) ( 𝑂 Func 𝑆 ) ( 2nd ‘ 𝐹 ) ) |
91 |
82
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
92 |
83
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑧 ∈ 𝐵 ) |
93 |
|
simpr2 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → 𝑤 ∈ 𝐵 ) |
94 |
93
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑤 ∈ 𝐵 ) |
95 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
96 |
95 41
|
eleqtrrdi |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑘 ∈ ( 𝑋 ( Hom ‘ 𝑂 ) 𝑧 ) ) |
97 |
|
simplr3 |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) |
98 |
28 29 87 88 90 91 92 94 96 97
|
funcco |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ( ℎ ( 〈 𝑋 , 𝑧 〉 ( comp ‘ 𝑂 ) 𝑤 ) 𝑘 ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ) ) |
99 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
100 |
2 99 4 91 92 94
|
oppcco |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ℎ ( 〈 𝑋 , 𝑧 〉 ( comp ‘ 𝑂 ) 𝑤 ) 𝑘 ) = ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) |
101 |
100
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ( ℎ ( 〈 𝑋 , 𝑧 〉 ( comp ‘ 𝑂 ) 𝑤 ) 𝑘 ) ) = ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) ) |
102 |
45
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → 𝑈 ∈ V ) |
103 |
102
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑈 ∈ V ) |
104 |
53
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ∈ 𝑈 ) |
105 |
55
|
3ad2antr1 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ∈ 𝑈 ) |
106 |
105
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ∈ 𝑈 ) |
107 |
52
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( 1st ‘ 𝐹 ) : 𝐵 ⟶ 𝑈 ) |
108 |
107 93
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ∈ 𝑈 ) |
109 |
108
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ∈ 𝑈 ) |
110 |
28 29 30 89 82 83
|
funcf2 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) : ( 𝑋 ( Hom ‘ 𝑂 ) 𝑧 ) ⟶ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
111 |
110
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) : ( 𝑋 ( Hom ‘ 𝑂 ) 𝑧 ) ⟶ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
112 |
111 96
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ∈ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
113 |
5 103 30 104 106
|
elsetchom |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ∈ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ↔ ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) : ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
114 |
112 113
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) : ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) |
115 |
28 29 30 89 83 93
|
funcf2 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) : ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ⟶ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ) |
116 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) |
117 |
115 116
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∈ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ) |
118 |
5 102 30 105 108
|
elsetchom |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∈ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ↔ ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) : ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ) |
119 |
117 118
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) : ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) |
120 |
119
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) : ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) |
121 |
5 103 88 104 106 109 114 120
|
setcco |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∘ ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ) ) |
122 |
98 101 121
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∘ ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ) ) |
123 |
122
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) ‘ 𝐴 ) = ( ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∘ ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ) ‘ 𝐴 ) ) |
124 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝐴 ∈ ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ) |
125 |
|
fvco3 |
⊢ ( ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) : ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ∧ 𝐴 ∈ ( ( 1st ‘ 𝐹 ) ‘ 𝑋 ) ) → ( ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∘ ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ) ‘ 𝐴 ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ‘ 𝐴 ) ) ) |
126 |
114 124 125
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∘ ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ) ‘ 𝐴 ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ‘ 𝐴 ) ) ) |
127 |
123 126
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) ‘ 𝐴 ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ‘ 𝐴 ) ) ) |
128 |
81
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝐶 ∈ Cat ) |
129 |
40 4
|
oppchom |
⊢ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) = ( 𝑤 ( Hom ‘ 𝐶 ) 𝑧 ) |
130 |
97 129
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ℎ ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑧 ) ) |
131 |
1 2 128 91 40 92 99 94 130 95
|
yon12 |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ‘ 𝑘 ) = ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) |
132 |
131
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ‘ ( ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ‘ 𝑘 ) ) = ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ‘ ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) ) |
133 |
13
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝑉 ∈ 𝑊 ) |
134 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ran ( Homf ‘ 𝐶 ) ⊆ 𝑈 ) |
135 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ran ( Homf ‘ 𝑄 ) ∪ 𝑈 ) ⊆ 𝑉 ) |
136 |
16
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → 𝐹 ∈ ( 𝑂 Func 𝑆 ) ) |
137 |
2 40 99 128 94 92 91 130 95
|
catcocl |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ∈ ( 𝑤 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
138 |
1 2 3 4 5 6 7 8 9 10 11 128 133 134 135 136 91 18 124 94 137
|
yonedalem4b |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ‘ ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) = ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) ‘ 𝐴 ) ) |
139 |
132 138
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ‘ ( ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ‘ 𝑘 ) ) = ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ( 𝑘 ( 〈 𝑤 , 𝑧 〉 ( comp ‘ 𝐶 ) 𝑋 ) ℎ ) ) ‘ 𝐴 ) ) |
140 |
1 2 3 4 5 6 7 8 9 10 11 128 133 134 135 136 91 18 124 92 95
|
yonedalem4b |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ‘ 𝑘 ) = ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ‘ 𝐴 ) ) |
141 |
140
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ‘ 𝑘 ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑘 ) ‘ 𝐴 ) ) ) |
142 |
127 139 141
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ‘ ( ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ‘ 𝑘 ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ‘ 𝑘 ) ) ) |
143 |
86 142
|
syldan |
⊢ ( ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) ∧ 𝑘 ∈ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ‘ ( ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ‘ 𝑘 ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ‘ 𝑘 ) ) ) |
144 |
143
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( 𝑘 ∈ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ↦ ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ‘ ( ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ‘ 𝑘 ) ) ) = ( 𝑘 ∈ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ↦ ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ‘ 𝑘 ) ) ) ) |
145 |
|
fveq2 |
⊢ ( 𝑧 = 𝑤 → ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) = ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ) |
146 |
|
fveq2 |
⊢ ( 𝑧 = 𝑤 → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) = ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ) |
147 |
|
fveq2 |
⊢ ( 𝑧 = 𝑤 → ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) = ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) |
148 |
145 146 147
|
feq123d |
⊢ ( 𝑧 = 𝑤 → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ↔ ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ) |
149 |
27
|
fveq1d |
⊢ ( 𝜑 → ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) = ( ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) ‘ 𝑧 ) ) |
150 |
|
ovex |
⊢ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ∈ V |
151 |
150
|
mptex |
⊢ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ∈ V |
152 |
|
eqid |
⊢ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) = ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) |
153 |
152
|
fvmpt2 |
⊢ ( ( 𝑧 ∈ 𝐵 ∧ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ∈ V ) → ( ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) ‘ 𝑧 ) = ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) |
154 |
151 153
|
mpan2 |
⊢ ( 𝑧 ∈ 𝐵 → ( ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) ‘ 𝑧 ) = ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) |
155 |
149 154
|
sylan9eq |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) = ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) ) |
156 |
155
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ↔ ( 𝑔 ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑋 ) ↦ ( ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑧 ) ‘ 𝑔 ) ‘ 𝐴 ) ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ) |
157 |
65 156
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐵 ) → ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) |
158 |
157
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) |
159 |
158
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ∀ 𝑧 ∈ 𝐵 ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) |
160 |
148 159 93
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) |
161 |
68
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ( 𝑂 Func 𝑆 ) ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ) |
162 |
28 29 30 161 83 93
|
funcf2 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) : ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ⟶ ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ) ) |
163 |
162 116
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ∈ ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ) ) |
164 |
72
|
3ad2antr1 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ∈ 𝑈 ) |
165 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) : 𝐵 ⟶ 𝑈 ) |
166 |
165 93
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ∈ 𝑈 ) |
167 |
5 102 30 164 166
|
elsetchom |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ∈ ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ) ↔ ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ) ) |
168 |
163 167
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ) |
169 |
|
fcompt |
⊢ ( ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ∧ ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ∘ ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ) = ( 𝑘 ∈ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ↦ ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ‘ ( ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ‘ 𝑘 ) ) ) ) |
170 |
160 168 169
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ∘ ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ) = ( 𝑘 ∈ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ↦ ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ‘ ( ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ‘ 𝑘 ) ) ) ) |
171 |
157
|
3ad2antr1 |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) |
172 |
|
fcompt |
⊢ ( ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) : ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ∧ ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) : ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ⟶ ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) → ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∘ ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ) = ( 𝑘 ∈ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ↦ ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ‘ 𝑘 ) ) ) ) |
173 |
119 171 172
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∘ ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ) = ( 𝑘 ∈ ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ↦ ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ‘ ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ‘ 𝑘 ) ) ) ) |
174 |
144 170 173
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ∘ ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∘ ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ) ) |
175 |
5 102 88 164 166 108 168 160
|
setcco |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) , ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ) = ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ∘ ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ) ) |
176 |
5 102 88 164 105 108 171 119
|
setcco |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ∘ ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ) ) |
177 |
174 175 176
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ) ) → ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) , ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ) ) |
178 |
177
|
ralrimivvva |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) , ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ) ) |
179 |
|
eqid |
⊢ ( 𝑂 Nat 𝑆 ) = ( 𝑂 Nat 𝑆 ) |
180 |
179 28 29 30 88 66 16
|
isnat2 |
⊢ ( 𝜑 → ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ∈ ( ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ( 𝑂 Nat 𝑆 ) 𝐹 ) ↔ ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ∈ X 𝑧 ∈ 𝐵 ( ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) ( Hom ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) ) ∧ ∀ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝑂 ) 𝑤 ) ( ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑤 ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) , ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑤 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( 𝑧 ( 2nd ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) 𝑤 ) ‘ ℎ ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐹 ) 𝑤 ) ‘ ℎ ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ) ‘ 𝑧 ) , ( ( 1st ‘ 𝐹 ) ‘ 𝑧 ) 〉 ( comp ‘ 𝑆 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑤 ) ) ( ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ‘ 𝑧 ) ) ) ) ) |
181 |
80 178 180
|
mpbir2and |
⊢ ( 𝜑 → ( ( 𝐹 𝑁 𝑋 ) ‘ 𝐴 ) ∈ ( ( ( 1st ‘ 𝑌 ) ‘ 𝑋 ) ( 𝑂 Nat 𝑆 ) 𝐹 ) ) |