Step | Hyp | Ref
| Expression |
1 | | ssid 3624 |
. . 3
⊢ 𝑅 ⊆ 𝑅 |
2 | 1 | a1i 11 |
. 2
⊢ (𝜑 → 𝑅 ⊆ 𝑅) |
3 | | eqid 2622 |
. . . . . . 7
⊢
(Base‘𝑎) =
(Base‘𝑎) |
4 | | eqid 2622 |
. . . . . . 7
⊢
(Base‘𝑏) =
(Base‘𝑏) |
5 | 3, 4 | rnghmf 41899 |
. . . . . 6
⊢ (ℎ ∈ (𝑎 RngHomo 𝑏) → ℎ:(Base‘𝑎)⟶(Base‘𝑏)) |
6 | | simpr 477 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ:(Base‘𝑎)⟶(Base‘𝑏)) |
7 | | fvex 6201 |
. . . . . . . . . 10
⊢
(Base‘𝑏)
∈ V |
8 | | fvex 6201 |
. . . . . . . . . 10
⊢
(Base‘𝑎)
∈ V |
9 | 7, 8 | pm3.2i 471 |
. . . . . . . . 9
⊢
((Base‘𝑏)
∈ V ∧ (Base‘𝑎) ∈ V) |
10 | | elmapg 7870 |
. . . . . . . . 9
⊢
(((Base‘𝑏)
∈ V ∧ (Base‘𝑎) ∈ V) → (ℎ ∈ ((Base‘𝑏) ↑𝑚
(Base‘𝑎)) ↔
ℎ:(Base‘𝑎)⟶(Base‘𝑏))) |
11 | 9, 10 | mp1i 13 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → (ℎ ∈ ((Base‘𝑏) ↑𝑚
(Base‘𝑎)) ↔
ℎ:(Base‘𝑎)⟶(Base‘𝑏))) |
12 | 6, 11 | mpbird 247 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑𝑚
(Base‘𝑎))) |
13 | 12 | ex 450 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ:(Base‘𝑎)⟶(Base‘𝑏) → ℎ ∈ ((Base‘𝑏) ↑𝑚
(Base‘𝑎)))) |
14 | 5, 13 | syl5 34 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ ∈ (𝑎 RngHomo 𝑏) → ℎ ∈ ((Base‘𝑏) ↑𝑚
(Base‘𝑎)))) |
15 | 14 | ssrdv 3609 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑𝑚
(Base‘𝑎))) |
16 | | ovres 6800 |
. . . . 5
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏)) |
17 | 16 | adantl 482 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏)) |
18 | | eqidd 2623 |
. . . . . 6
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥)))) |
19 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏)) |
20 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎)) |
21 | 19, 20 | oveqan12rd 6670 |
. . . . . . 7
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚
(Base‘𝑥)) =
((Base‘𝑏)
↑𝑚 (Base‘𝑎))) |
22 | 21 | adantl 482 |
. . . . . 6
⊢ (((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → ((Base‘𝑦) ↑𝑚
(Base‘𝑥)) =
((Base‘𝑏)
↑𝑚 (Base‘𝑎))) |
23 | | simpl 473 |
. . . . . 6
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → 𝑎 ∈ 𝑅) |
24 | | simpr 477 |
. . . . . 6
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑅) |
25 | | ovexd 6680 |
. . . . . 6
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → ((Base‘𝑏) ↑𝑚
(Base‘𝑎)) ∈
V) |
26 | 18, 22, 23, 24, 25 | ovmpt2d 6788 |
. . . . 5
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚
(Base‘𝑎))) |
27 | 26 | adantl 482 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚
(Base‘𝑎))) |
28 | 15, 17, 27 | 3sstr4d 3648 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥)))𝑏)) |
29 | 28 | ralrimivva 2971 |
. 2
⊢ (𝜑 → ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥)))𝑏)) |
30 | | rnghmfn 41890 |
. . . . 5
⊢ RngHomo
Fn (Rng × Rng) |
31 | 30 | a1i 11 |
. . . 4
⊢ (𝜑 → RngHomo Fn (Rng ×
Rng)) |
32 | | rnghmsscmap.r |
. . . . . 6
⊢ (𝜑 → 𝑅 = (Rng ∩ 𝑈)) |
33 | | inss1 3833 |
. . . . . 6
⊢ (Rng
∩ 𝑈) ⊆
Rng |
34 | 32, 33 | syl6eqss 3655 |
. . . . 5
⊢ (𝜑 → 𝑅 ⊆ Rng) |
35 | | xpss12 5225 |
. . . . 5
⊢ ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng ×
Rng)) |
36 | 34, 34, 35 | syl2anc 693 |
. . . 4
⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Rng ×
Rng)) |
37 | | fnssres 6004 |
. . . 4
⊢ ((
RngHomo Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → (
RngHomo ↾ (𝑅 ×
𝑅)) Fn (𝑅 × 𝑅)) |
38 | 31, 36, 37 | syl2anc 693 |
. . 3
⊢ (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅)) |
39 | | eqid 2622 |
. . . . 5
⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥))) |
40 | | ovex 6678 |
. . . . 5
⊢
((Base‘𝑦)
↑𝑚 (Base‘𝑥)) ∈ V |
41 | 39, 40 | fnmpt2i 7239 |
. . . 4
⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥))) Fn (𝑅 × 𝑅) |
42 | 41 | a1i 11 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥))) Fn (𝑅 × 𝑅)) |
43 | | incom 3805 |
. . . . 5
⊢ (Rng
∩ 𝑈) = (𝑈 ∩ Rng) |
44 | | rnghmsscmap.u |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ 𝑉) |
45 | | inex1g 4801 |
. . . . . 6
⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) |
46 | 44, 45 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
47 | 43, 46 | syl5eqel 2705 |
. . . 4
⊢ (𝜑 → (Rng ∩ 𝑈) ∈ V) |
48 | 32, 47 | eqeltrd 2701 |
. . 3
⊢ (𝜑 → 𝑅 ∈ V) |
49 | 38, 42, 48 | isssc 16480 |
. 2
⊢ (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)))) |
50 | 2, 29, 49 | mpbir2and 957 |
1
⊢ (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥)))) |