Step |
Hyp |
Ref |
Expression |
1 |
|
rnghmsscmap.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
2 |
|
rnghmsscmap.r |
⊢ ( 𝜑 → 𝑅 = ( Rng ∩ 𝑈 ) ) |
3 |
|
ssidd |
⊢ ( 𝜑 → 𝑅 ⊆ 𝑅 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝑎 ) = ( Base ‘ 𝑎 ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝑏 ) = ( Base ‘ 𝑏 ) |
6 |
4 5
|
rnghmf |
⊢ ( ℎ ∈ ( 𝑎 RngHomo 𝑏 ) → ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) |
7 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) |
8 |
|
fvex |
⊢ ( Base ‘ 𝑏 ) ∈ V |
9 |
|
fvex |
⊢ ( Base ‘ 𝑎 ) ∈ V |
10 |
8 9
|
pm3.2i |
⊢ ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V ) |
11 |
|
elmapg |
⊢ ( ( ( Base ‘ 𝑏 ) ∈ V ∧ ( Base ‘ 𝑎 ) ∈ V ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) ) |
12 |
10 11
|
mp1i |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ( ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ↔ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) ) |
13 |
7 12
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) ∧ ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
14 |
13
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ℎ : ( Base ‘ 𝑎 ) ⟶ ( Base ‘ 𝑏 ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) ) |
15 |
6 14
|
syl5 |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( ℎ ∈ ( 𝑎 RngHomo 𝑏 ) → ℎ ∈ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) ) |
16 |
15
|
ssrdv |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 RngHomo 𝑏 ) ⊆ ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
17 |
|
ovres |
⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) = ( 𝑎 RngHomo 𝑏 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) = ( 𝑎 RngHomo 𝑏 ) ) |
19 |
|
eqidd |
⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) |
20 |
|
fveq2 |
⊢ ( 𝑦 = 𝑏 → ( Base ‘ 𝑦 ) = ( Base ‘ 𝑏 ) ) |
21 |
|
fveq2 |
⊢ ( 𝑥 = 𝑎 → ( Base ‘ 𝑥 ) = ( Base ‘ 𝑎 ) ) |
22 |
20 21
|
oveqan12rd |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
23 |
22
|
adantl |
⊢ ( ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ∧ ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ) → ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
24 |
|
simpl |
⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → 𝑎 ∈ 𝑅 ) |
25 |
|
simpr |
⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → 𝑏 ∈ 𝑅 ) |
26 |
|
ovexd |
⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ∈ V ) |
27 |
19 23 24 25 26
|
ovmpod |
⊢ ( ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) → ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
28 |
27
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) = ( ( Base ‘ 𝑏 ) ↑m ( Base ‘ 𝑎 ) ) ) |
29 |
16 18 28
|
3sstr4d |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅 ) ) → ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) ) |
30 |
29
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑅 ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) ) |
31 |
|
rnghmfn |
⊢ RngHomo Fn ( Rng × Rng ) |
32 |
31
|
a1i |
⊢ ( 𝜑 → RngHomo Fn ( Rng × Rng ) ) |
33 |
|
inss1 |
⊢ ( Rng ∩ 𝑈 ) ⊆ Rng |
34 |
2 33
|
eqsstrdi |
⊢ ( 𝜑 → 𝑅 ⊆ Rng ) |
35 |
|
xpss12 |
⊢ ( ( 𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng ) → ( 𝑅 × 𝑅 ) ⊆ ( Rng × Rng ) ) |
36 |
34 34 35
|
syl2anc |
⊢ ( 𝜑 → ( 𝑅 × 𝑅 ) ⊆ ( Rng × Rng ) ) |
37 |
|
fnssres |
⊢ ( ( RngHomo Fn ( Rng × Rng ) ∧ ( 𝑅 × 𝑅 ) ⊆ ( Rng × Rng ) ) → ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) ) |
38 |
32 36 37
|
syl2anc |
⊢ ( 𝜑 → ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) ) |
39 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) |
40 |
|
ovex |
⊢ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ∈ V |
41 |
39 40
|
fnmpoi |
⊢ ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑅 × 𝑅 ) |
42 |
41
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) Fn ( 𝑅 × 𝑅 ) ) |
43 |
|
incom |
⊢ ( Rng ∩ 𝑈 ) = ( 𝑈 ∩ Rng ) |
44 |
|
inex1g |
⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ Rng ) ∈ V ) |
45 |
1 44
|
syl |
⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) ∈ V ) |
46 |
43 45
|
eqeltrid |
⊢ ( 𝜑 → ( Rng ∩ 𝑈 ) ∈ V ) |
47 |
2 46
|
eqeltrd |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
48 |
38 42 47
|
isssc |
⊢ ( 𝜑 → ( ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ↔ ( 𝑅 ⊆ 𝑅 ∧ ∀ 𝑎 ∈ 𝑅 ∀ 𝑏 ∈ 𝑅 ( 𝑎 ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) 𝑏 ) ⊆ ( 𝑎 ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) 𝑏 ) ) ) ) |
49 |
3 30 48
|
mpbir2and |
⊢ ( 𝜑 → ( RngHomo ↾ ( 𝑅 × 𝑅 ) ) ⊆cat ( 𝑥 ∈ 𝑅 , 𝑦 ∈ 𝑅 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ) |