Step |
Hyp |
Ref |
Expression |
1 |
|
dmrelrnrel.x |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
dmrelrnrel.y |
⊢ Ⅎ 𝑦 𝜑 |
3 |
|
dmrelrnrel.i |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) |
4 |
|
dmrelrnrel.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝐴 ) |
5 |
|
dmrelrnrel.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐴 ) |
6 |
|
dmrelrnrel.r |
⊢ ( 𝜑 → 𝐵 𝑅 𝐶 ) |
7 |
|
id |
⊢ ( 𝜑 → 𝜑 ) |
8 |
7 4 5
|
jca31 |
⊢ ( 𝜑 → ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐴 ) ) |
9 |
|
nfv |
⊢ Ⅎ 𝑦 𝐵 ∈ 𝐴 |
10 |
2 9
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) |
11 |
|
nfv |
⊢ Ⅎ 𝑦 𝐶 ∈ 𝐴 |
12 |
10 11
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐴 ) |
13 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝐵 𝑅 𝐶 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) |
14 |
12 13
|
nfim |
⊢ Ⅎ 𝑦 ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐴 ) → ( 𝐵 𝑅 𝐶 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) ) |
15 |
9 14
|
nfim |
⊢ Ⅎ 𝑦 ( 𝐵 ∈ 𝐴 → ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐴 ) → ( 𝐵 𝑅 𝐶 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) ) ) |
16 |
|
eleq1 |
⊢ ( 𝑦 = 𝐶 → ( 𝑦 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴 ) ) |
17 |
16
|
anbi2d |
⊢ ( 𝑦 = 𝐶 → ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ↔ ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐴 ) ) ) |
18 |
|
breq2 |
⊢ ( 𝑦 = 𝐶 → ( 𝐵 𝑅 𝑦 ↔ 𝐵 𝑅 𝐶 ) ) |
19 |
|
fveq2 |
⊢ ( 𝑦 = 𝐶 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐶 ) ) |
20 |
19
|
breq2d |
⊢ ( 𝑦 = 𝐶 → ( ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) ) |
21 |
18 20
|
imbi12d |
⊢ ( 𝑦 = 𝐶 → ( ( 𝐵 𝑅 𝑦 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐵 𝑅 𝐶 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) ) ) |
22 |
17 21
|
imbi12d |
⊢ ( 𝑦 = 𝐶 → ( ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐵 𝑅 𝑦 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) ↔ ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐴 ) → ( 𝐵 𝑅 𝐶 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) ) ) ) |
23 |
22
|
imbi2d |
⊢ ( 𝑦 = 𝐶 → ( ( 𝐵 ∈ 𝐴 → ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐵 𝑅 𝑦 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) ) ↔ ( 𝐵 ∈ 𝐴 → ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐴 ) → ( 𝐵 𝑅 𝐶 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) ) ) ) ) |
24 |
|
nfv |
⊢ Ⅎ 𝑥 𝐵 ∈ 𝐴 |
25 |
1 24
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) |
26 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴 |
27 |
25 26
|
nfan |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) |
28 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝐵 𝑅 𝑦 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) |
29 |
27 28
|
nfim |
⊢ Ⅎ 𝑥 ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐵 𝑅 𝑦 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) |
30 |
|
eleq1 |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴 ) ) |
31 |
30
|
anbi2d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ) ) |
32 |
31
|
anbi1d |
⊢ ( 𝑥 = 𝐵 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ↔ ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) ) ) |
33 |
|
breq1 |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 𝑅 𝑦 ↔ 𝐵 𝑅 𝑦 ) ) |
34 |
|
fveq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐵 ) ) |
35 |
34
|
breq1d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) |
36 |
33 35
|
imbi12d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 𝑅 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐵 𝑅 𝑦 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) ) |
37 |
32 36
|
imbi12d |
⊢ ( 𝑥 = 𝐵 → ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) ↔ ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐵 𝑅 𝑦 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
38 |
3
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) |
39 |
38
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 → ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) |
40 |
29 37 39
|
vtoclg1f |
⊢ ( 𝐵 ∈ 𝐴 → ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐵 𝑅 𝑦 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) ) |
41 |
15 23 40
|
vtoclg1f |
⊢ ( 𝐶 ∈ 𝐴 → ( 𝐵 ∈ 𝐴 → ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐴 ) → ( 𝐵 𝑅 𝐶 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) ) ) ) |
42 |
5 4 41
|
sylc |
⊢ ( 𝜑 → ( ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝐶 ∈ 𝐴 ) → ( 𝐵 𝑅 𝐶 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) ) ) |
43 |
8 6 42
|
mp2d |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) 𝑆 ( 𝐹 ‘ 𝐶 ) ) |