Step |
Hyp |
Ref |
Expression |
1 |
|
prf1st.p |
⊢ 𝑃 = ( 𝐹 〈,〉F 𝐺 ) |
2 |
|
prf1st.c |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) |
3 |
|
prf1st.d |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝐶 Func 𝐸 ) ) |
4 |
|
eqid |
⊢ ( 𝐷 ×c 𝐸 ) = ( 𝐷 ×c 𝐸 ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 ) |
7 |
4 5 6
|
xpcbas |
⊢ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) = ( Base ‘ ( 𝐷 ×c 𝐸 ) ) |
8 |
|
eqid |
⊢ ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) = ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) |
9 |
|
funcrcl |
⊢ ( 𝐹 ∈ ( 𝐶 Func 𝐷 ) → ( 𝐶 ∈ Cat ∧ 𝐷 ∈ Cat ) ) |
10 |
2 9
|
syl |
⊢ ( 𝜑 → ( 𝐶 ∈ Cat ∧ 𝐷 ∈ Cat ) ) |
11 |
10
|
simprd |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → 𝐷 ∈ Cat ) |
13 |
|
funcrcl |
⊢ ( 𝐺 ∈ ( 𝐶 Func 𝐸 ) → ( 𝐶 ∈ Cat ∧ 𝐸 ∈ Cat ) ) |
14 |
3 13
|
syl |
⊢ ( 𝜑 → ( 𝐶 ∈ Cat ∧ 𝐸 ∈ Cat ) ) |
15 |
14
|
simprd |
⊢ ( 𝜑 → 𝐸 ∈ Cat ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → 𝐸 ∈ Cat ) |
17 |
|
eqid |
⊢ ( 𝐷 1stF 𝐸 ) = ( 𝐷 1stF 𝐸 ) |
18 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
19 |
|
relfunc |
⊢ Rel ( 𝐶 Func 𝐷 ) |
20 |
|
1st2ndbr |
⊢ ( ( Rel ( 𝐶 Func 𝐷 ) ∧ 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) → ( 1st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) ( 2nd ‘ 𝐹 ) ) |
21 |
19 2 20
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) ( 2nd ‘ 𝐹 ) ) |
22 |
18 5 21
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) : ( Base ‘ 𝐶 ) ⟶ ( Base ‘ 𝐷 ) ) |
23 |
22
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐷 ) ) |
24 |
|
relfunc |
⊢ Rel ( 𝐶 Func 𝐸 ) |
25 |
|
1st2ndbr |
⊢ ( ( Rel ( 𝐶 Func 𝐸 ) ∧ 𝐺 ∈ ( 𝐶 Func 𝐸 ) ) → ( 1st ‘ 𝐺 ) ( 𝐶 Func 𝐸 ) ( 2nd ‘ 𝐺 ) ) |
26 |
24 3 25
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ 𝐺 ) ( 𝐶 Func 𝐸 ) ( 2nd ‘ 𝐺 ) ) |
27 |
18 6 26
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ 𝐺 ) : ( Base ‘ 𝐶 ) ⟶ ( Base ‘ 𝐸 ) ) |
28 |
27
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐸 ) ) |
29 |
23 28
|
opelxpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ∈ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ) |
30 |
4 7 8 12 16 17 29
|
1stf1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) = ( 1st ‘ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) ) |
31 |
|
fvex |
⊢ ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) ∈ V |
32 |
|
fvex |
⊢ ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) ∈ V |
33 |
31 32
|
op1st |
⊢ ( 1st ‘ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) = ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) |
34 |
30 33
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) = ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) ) |
35 |
34
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) ) ) |
36 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
37 |
1 18 36 2 3
|
prfval |
⊢ ( 𝜑 → 𝑃 = 〈 ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( ℎ ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 ) |
38 |
|
fvex |
⊢ ( Base ‘ 𝐶 ) ∈ V |
39 |
38
|
mptex |
⊢ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) ∈ V |
40 |
38 38
|
mpoex |
⊢ ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( ℎ ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) ∈ V |
41 |
39 40
|
op1std |
⊢ ( 𝑃 = 〈 ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( ℎ ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 → ( 1st ‘ 𝑃 ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) ) |
42 |
37 41
|
syl |
⊢ ( 𝜑 → ( 1st ‘ 𝑃 ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) ) |
43 |
|
relfunc |
⊢ Rel ( ( 𝐷 ×c 𝐸 ) Func 𝐷 ) |
44 |
4 11 15 17
|
1stfcl |
⊢ ( 𝜑 → ( 𝐷 1stF 𝐸 ) ∈ ( ( 𝐷 ×c 𝐸 ) Func 𝐷 ) ) |
45 |
|
1st2ndbr |
⊢ ( ( Rel ( ( 𝐷 ×c 𝐸 ) Func 𝐷 ) ∧ ( 𝐷 1stF 𝐸 ) ∈ ( ( 𝐷 ×c 𝐸 ) Func 𝐷 ) ) → ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 𝐷 ×c 𝐸 ) Func 𝐷 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ) |
46 |
43 44 45
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 𝐷 ×c 𝐸 ) Func 𝐷 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ) |
47 |
7 5 46
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) : ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ⟶ ( Base ‘ 𝐷 ) ) |
48 |
47
|
feqmptd |
⊢ ( 𝜑 → ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) = ( 𝑢 ∈ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ↦ ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ 𝑢 ) ) ) |
49 |
|
fveq2 |
⊢ ( 𝑢 = 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 → ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ 𝑢 ) = ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) ) |
50 |
29 42 48 49
|
fmptco |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ∘ ( 1st ‘ 𝑃 ) ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) ) ) |
51 |
22
|
feqmptd |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) ) ) |
52 |
35 50 51
|
3eqtr4d |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ∘ ( 1st ‘ 𝑃 ) ) = ( 1st ‘ 𝐹 ) ) |
53 |
11
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝐷 ∈ Cat ) |
54 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝐸 ∈ Cat ) |
55 |
|
relfunc |
⊢ Rel ( 𝐶 Func ( 𝐷 ×c 𝐸 ) ) |
56 |
1 4 2 3
|
prfcl |
⊢ ( 𝜑 → 𝑃 ∈ ( 𝐶 Func ( 𝐷 ×c 𝐸 ) ) ) |
57 |
|
1st2ndbr |
⊢ ( ( Rel ( 𝐶 Func ( 𝐷 ×c 𝐸 ) ) ∧ 𝑃 ∈ ( 𝐶 Func ( 𝐷 ×c 𝐸 ) ) ) → ( 1st ‘ 𝑃 ) ( 𝐶 Func ( 𝐷 ×c 𝐸 ) ) ( 2nd ‘ 𝑃 ) ) |
58 |
55 56 57
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ 𝑃 ) ( 𝐶 Func ( 𝐷 ×c 𝐸 ) ) ( 2nd ‘ 𝑃 ) ) |
59 |
18 7 58
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ 𝑃 ) : ( Base ‘ 𝐶 ) ⟶ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ) |
60 |
59
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ∈ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ) |
61 |
60
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ∈ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ) |
62 |
61
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ∈ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ) |
63 |
59
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ∈ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ) |
64 |
63
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ∈ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ) |
65 |
64
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ∈ ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐸 ) ) ) |
66 |
4 7 8 53 54 17 62 65
|
1stf2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) = ( 1st ↾ ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ) ) |
67 |
66
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) = ( ( 1st ↾ ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ) ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) ) |
68 |
58
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 1st ‘ 𝑃 ) ( 𝐶 Func ( 𝐷 ×c 𝐸 ) ) ( 2nd ‘ 𝑃 ) ) |
69 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
70 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) ) |
71 |
18 36 8 68 69 70
|
funcf2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) : ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ⟶ ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ) |
72 |
71
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ∈ ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ) |
73 |
72
|
fvresd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( 1st ↾ ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ) ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) = ( 1st ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) ) |
74 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) |
75 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝐺 ∈ ( 𝐶 Func 𝐸 ) ) |
76 |
69
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
77 |
70
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) ) |
78 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) |
79 |
1 18 36 74 75 76 77 78
|
prf2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) = 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑓 ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ 𝑓 ) 〉 ) |
80 |
79
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( 1st ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) = ( 1st ‘ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑓 ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ 𝑓 ) 〉 ) ) |
81 |
|
fvex |
⊢ ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑓 ) ∈ V |
82 |
|
fvex |
⊢ ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ 𝑓 ) ∈ V |
83 |
81 82
|
op1st |
⊢ ( 1st ‘ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑓 ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ 𝑓 ) 〉 ) = ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑓 ) |
84 |
80 83
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( 1st ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) = ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑓 ) ) |
85 |
67 73 84
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) = ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑓 ) ) |
86 |
85
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↦ ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) ) = ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↦ ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑓 ) ) ) |
87 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
88 |
46
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 𝐷 ×c 𝐸 ) Func 𝐷 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ) |
89 |
7 8 87 88 61 64
|
funcf2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) : ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ⟶ ( ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ) ( Hom ‘ 𝐷 ) ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ) ) |
90 |
|
fcompt |
⊢ ( ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) : ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ⟶ ( ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ) ( Hom ‘ 𝐷 ) ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ‘ ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ) ∧ ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) : ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ⟶ ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ×c 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ) → ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ∘ ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ) = ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↦ ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) ) ) |
91 |
89 71 90
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ∘ ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ) = ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↦ ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ‘ ( ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ‘ 𝑓 ) ) ) ) |
92 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 1st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) ( 2nd ‘ 𝐹 ) ) |
93 |
18 36 87 92 69 70
|
funcf2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) : ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ⟶ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑦 ) ) ) |
94 |
93
|
feqmptd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) = ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↦ ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ 𝑓 ) ) ) |
95 |
86 91 94
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ∘ ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ) = ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ) |
96 |
95
|
3impb |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ∘ ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ) = ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ) |
97 |
96
|
mpoeq3dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ∘ ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ) ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ) ) |
98 |
18 21
|
funcfn2 |
⊢ ( 𝜑 → ( 2nd ‘ 𝐹 ) Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) |
99 |
|
fnov |
⊢ ( ( 2nd ‘ 𝐹 ) Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ↔ ( 2nd ‘ 𝐹 ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ) ) |
100 |
98 99
|
sylib |
⊢ ( 𝜑 → ( 2nd ‘ 𝐹 ) = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ) ) |
101 |
97 100
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ∘ ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ) ) = ( 2nd ‘ 𝐹 ) ) |
102 |
52 101
|
opeq12d |
⊢ ( 𝜑 → 〈 ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ∘ ( 1st ‘ 𝑃 ) ) , ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ∘ ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ) ) 〉 = 〈 ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) 〉 ) |
103 |
18 56 44
|
cofuval |
⊢ ( 𝜑 → ( ( 𝐷 1stF 𝐸 ) ∘func 𝑃 ) = 〈 ( ( 1st ‘ ( 𝐷 1stF 𝐸 ) ) ∘ ( 1st ‘ 𝑃 ) ) , ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( ( ( ( 1st ‘ 𝑃 ) ‘ 𝑥 ) ( 2nd ‘ ( 𝐷 1stF 𝐸 ) ) ( ( 1st ‘ 𝑃 ) ‘ 𝑦 ) ) ∘ ( 𝑥 ( 2nd ‘ 𝑃 ) 𝑦 ) ) ) 〉 ) |
104 |
|
1st2nd |
⊢ ( ( Rel ( 𝐶 Func 𝐷 ) ∧ 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) → 𝐹 = 〈 ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) 〉 ) |
105 |
19 2 104
|
sylancr |
⊢ ( 𝜑 → 𝐹 = 〈 ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) 〉 ) |
106 |
102 103 105
|
3eqtr4d |
⊢ ( 𝜑 → ( ( 𝐷 1stF 𝐸 ) ∘func 𝑃 ) = 𝐹 ) |