Step |
Hyp |
Ref |
Expression |
1 |
|
resf1st.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
2 |
|
resf1st.h |
⊢ ( 𝜑 → 𝐻 ∈ 𝑊 ) |
3 |
|
resf1st.s |
⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) ) |
4 |
|
resf2nd.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑆 ) |
5 |
|
resf2nd.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑆 ) |
6 |
|
df-ov |
⊢ ( 𝑋 ( 2nd ‘ ( 𝐹 ↾f 𝐻 ) ) 𝑌 ) = ( ( 2nd ‘ ( 𝐹 ↾f 𝐻 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) |
7 |
1 2
|
resfval |
⊢ ( 𝜑 → ( 𝐹 ↾f 𝐻 ) = 〈 ( ( 1st ‘ 𝐹 ) ↾ dom dom 𝐻 ) , ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) 〉 ) |
8 |
7
|
fveq2d |
⊢ ( 𝜑 → ( 2nd ‘ ( 𝐹 ↾f 𝐻 ) ) = ( 2nd ‘ 〈 ( ( 1st ‘ 𝐹 ) ↾ dom dom 𝐻 ) , ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) 〉 ) ) |
9 |
|
fvex |
⊢ ( 1st ‘ 𝐹 ) ∈ V |
10 |
9
|
resex |
⊢ ( ( 1st ‘ 𝐹 ) ↾ dom dom 𝐻 ) ∈ V |
11 |
|
dmexg |
⊢ ( 𝐻 ∈ 𝑊 → dom 𝐻 ∈ V ) |
12 |
|
mptexg |
⊢ ( dom 𝐻 ∈ V → ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) ∈ V ) |
13 |
2 11 12
|
3syl |
⊢ ( 𝜑 → ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) ∈ V ) |
14 |
|
op2ndg |
⊢ ( ( ( ( 1st ‘ 𝐹 ) ↾ dom dom 𝐻 ) ∈ V ∧ ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) ∈ V ) → ( 2nd ‘ 〈 ( ( 1st ‘ 𝐹 ) ↾ dom dom 𝐻 ) , ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) 〉 ) = ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) ) |
15 |
10 13 14
|
sylancr |
⊢ ( 𝜑 → ( 2nd ‘ 〈 ( ( 1st ‘ 𝐹 ) ↾ dom dom 𝐻 ) , ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) 〉 ) = ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) ) |
16 |
8 15
|
eqtrd |
⊢ ( 𝜑 → ( 2nd ‘ ( 𝐹 ↾f 𝐻 ) ) = ( 𝑧 ∈ dom 𝐻 ↦ ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) ) ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 = 〈 𝑋 , 𝑌 〉 ) → 𝑧 = 〈 𝑋 , 𝑌 〉 ) |
18 |
17
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑧 = 〈 𝑋 , 𝑌 〉 ) → ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) = ( ( 2nd ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑌 〉 ) ) |
19 |
|
df-ov |
⊢ ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) = ( ( 2nd ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑌 〉 ) |
20 |
18 19
|
eqtr4di |
⊢ ( ( 𝜑 ∧ 𝑧 = 〈 𝑋 , 𝑌 〉 ) → ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) = ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ) |
21 |
17
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑧 = 〈 𝑋 , 𝑌 〉 ) → ( 𝐻 ‘ 𝑧 ) = ( 𝐻 ‘ 〈 𝑋 , 𝑌 〉 ) ) |
22 |
|
df-ov |
⊢ ( 𝑋 𝐻 𝑌 ) = ( 𝐻 ‘ 〈 𝑋 , 𝑌 〉 ) |
23 |
21 22
|
eqtr4di |
⊢ ( ( 𝜑 ∧ 𝑧 = 〈 𝑋 , 𝑌 〉 ) → ( 𝐻 ‘ 𝑧 ) = ( 𝑋 𝐻 𝑌 ) ) |
24 |
20 23
|
reseq12d |
⊢ ( ( 𝜑 ∧ 𝑧 = 〈 𝑋 , 𝑌 〉 ) → ( ( ( 2nd ‘ 𝐹 ) ‘ 𝑧 ) ↾ ( 𝐻 ‘ 𝑧 ) ) = ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ↾ ( 𝑋 𝐻 𝑌 ) ) ) |
25 |
4 5
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( 𝑆 × 𝑆 ) ) |
26 |
3
|
fndmd |
⊢ ( 𝜑 → dom 𝐻 = ( 𝑆 × 𝑆 ) ) |
27 |
25 26
|
eleqtrrd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ dom 𝐻 ) |
28 |
|
ovex |
⊢ ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ∈ V |
29 |
28
|
resex |
⊢ ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ↾ ( 𝑋 𝐻 𝑌 ) ) ∈ V |
30 |
29
|
a1i |
⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ↾ ( 𝑋 𝐻 𝑌 ) ) ∈ V ) |
31 |
16 24 27 30
|
fvmptd |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 𝐹 ↾f 𝐻 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ↾ ( 𝑋 𝐻 𝑌 ) ) ) |
32 |
6 31
|
eqtrid |
⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ ( 𝐹 ↾f 𝐻 ) ) 𝑌 ) = ( ( 𝑋 ( 2nd ‘ 𝐹 ) 𝑌 ) ↾ ( 𝑋 𝐻 𝑌 ) ) ) |