Step |
Hyp |
Ref |
Expression |
1 |
|
fcores.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
2 |
|
fcores.e |
⊢ 𝐸 = ( ran 𝐹 ∩ 𝐶 ) |
3 |
|
fcores.p |
⊢ 𝑃 = ( ◡ 𝐹 “ 𝐶 ) |
4 |
|
fcores.x |
⊢ 𝑋 = ( 𝐹 ↾ 𝑃 ) |
5 |
|
fcores.g |
⊢ ( 𝜑 → 𝐺 : 𝐶 ⟶ 𝐷 ) |
6 |
|
fcores.y |
⊢ 𝑌 = ( 𝐺 ↾ 𝐸 ) |
7 |
|
f1cof1blem.s |
⊢ ( 𝜑 → ran 𝐹 = 𝐶 ) |
8 |
7
|
eqcomd |
⊢ ( 𝜑 → 𝐶 = ran 𝐹 ) |
9 |
8
|
imaeq2d |
⊢ ( 𝜑 → ( ◡ 𝐹 “ 𝐶 ) = ( ◡ 𝐹 “ ran 𝐹 ) ) |
10 |
3 9
|
syl5eq |
⊢ ( 𝜑 → 𝑃 = ( ◡ 𝐹 “ ran 𝐹 ) ) |
11 |
|
cnvimarndm |
⊢ ( ◡ 𝐹 “ ran 𝐹 ) = dom 𝐹 |
12 |
1
|
fdmd |
⊢ ( 𝜑 → dom 𝐹 = 𝐴 ) |
13 |
11 12
|
syl5eq |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ran 𝐹 ) = 𝐴 ) |
14 |
10 13
|
eqtrd |
⊢ ( 𝜑 → 𝑃 = 𝐴 ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ ran 𝐹 = 𝐶 ) → ran 𝐹 = 𝐶 ) |
16 |
15
|
ineq1d |
⊢ ( ( 𝜑 ∧ ran 𝐹 = 𝐶 ) → ( ran 𝐹 ∩ 𝐶 ) = ( 𝐶 ∩ 𝐶 ) ) |
17 |
|
inidm |
⊢ ( 𝐶 ∩ 𝐶 ) = 𝐶 |
18 |
16 17
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ran 𝐹 = 𝐶 ) → ( ran 𝐹 ∩ 𝐶 ) = 𝐶 ) |
19 |
7 18
|
mpdan |
⊢ ( 𝜑 → ( ran 𝐹 ∩ 𝐶 ) = 𝐶 ) |
20 |
2 19
|
syl5eq |
⊢ ( 𝜑 → 𝐸 = 𝐶 ) |
21 |
14 20
|
jca |
⊢ ( 𝜑 → ( 𝑃 = 𝐴 ∧ 𝐸 = 𝐶 ) ) |
22 |
10 11
|
eqtrdi |
⊢ ( 𝜑 → 𝑃 = dom 𝐹 ) |
23 |
22
|
reseq2d |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝑃 ) = ( 𝐹 ↾ dom 𝐹 ) ) |
24 |
4 23
|
syl5eq |
⊢ ( 𝜑 → 𝑋 = ( 𝐹 ↾ dom 𝐹 ) ) |
25 |
1
|
freld |
⊢ ( 𝜑 → Rel 𝐹 ) |
26 |
|
resdm |
⊢ ( Rel 𝐹 → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 ) |
27 |
25 26
|
syl |
⊢ ( 𝜑 → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 ) |
28 |
24 27
|
eqtrd |
⊢ ( 𝜑 → 𝑋 = 𝐹 ) |
29 |
5
|
fdmd |
⊢ ( 𝜑 → dom 𝐺 = 𝐶 ) |
30 |
20 29
|
eqtr4d |
⊢ ( 𝜑 → 𝐸 = dom 𝐺 ) |
31 |
30
|
reseq2d |
⊢ ( 𝜑 → ( 𝐺 ↾ 𝐸 ) = ( 𝐺 ↾ dom 𝐺 ) ) |
32 |
6 31
|
syl5eq |
⊢ ( 𝜑 → 𝑌 = ( 𝐺 ↾ dom 𝐺 ) ) |
33 |
5
|
freld |
⊢ ( 𝜑 → Rel 𝐺 ) |
34 |
|
resdm |
⊢ ( Rel 𝐺 → ( 𝐺 ↾ dom 𝐺 ) = 𝐺 ) |
35 |
33 34
|
syl |
⊢ ( 𝜑 → ( 𝐺 ↾ dom 𝐺 ) = 𝐺 ) |
36 |
32 35
|
eqtrd |
⊢ ( 𝜑 → 𝑌 = 𝐺 ) |
37 |
21 28 36
|
jca32 |
⊢ ( 𝜑 → ( ( 𝑃 = 𝐴 ∧ 𝐸 = 𝐶 ) ∧ ( 𝑋 = 𝐹 ∧ 𝑌 = 𝐺 ) ) ) |