| Step | Hyp | Ref
| Expression |
| 1 | | ssidd 4007 |
. 2
⊢ (𝜑 → 𝑅 ⊆ 𝑅) |
| 2 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝑎) =
(Base‘𝑎) |
| 3 | | eqid 2737 |
. . . . . . 7
⊢
(Base‘𝑏) =
(Base‘𝑏) |
| 4 | 2, 3 | rnghmf 20448 |
. . . . . 6
⊢ (ℎ ∈ (𝑎 RngHom 𝑏) → ℎ:(Base‘𝑎)⟶(Base‘𝑏)) |
| 5 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ:(Base‘𝑎)⟶(Base‘𝑏)) |
| 6 | | fvex 6919 |
. . . . . . . . . 10
⊢
(Base‘𝑏)
∈ V |
| 7 | | fvex 6919 |
. . . . . . . . . 10
⊢
(Base‘𝑎)
∈ V |
| 8 | 6, 7 | pm3.2i 470 |
. . . . . . . . 9
⊢
((Base‘𝑏)
∈ V ∧ (Base‘𝑎) ∈ V) |
| 9 | | elmapg 8879 |
. . . . . . . . 9
⊢
(((Base‘𝑏)
∈ V ∧ (Base‘𝑎) ∈ V) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏))) |
| 10 | 8, 9 | mp1i 13 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → (ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ ℎ:(Base‘𝑎)⟶(Base‘𝑏))) |
| 11 | 5, 10 | mpbird 257 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) ∧ ℎ:(Base‘𝑎)⟶(Base‘𝑏)) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎))) |
| 12 | 11 | ex 412 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ:(Base‘𝑎)⟶(Base‘𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))) |
| 13 | 4, 12 | syl5 34 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (ℎ ∈ (𝑎 RngHom 𝑏) → ℎ ∈ ((Base‘𝑏) ↑m (Base‘𝑎)))) |
| 14 | 13 | ssrdv 3989 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎))) |
| 15 | | ovres 7599 |
. . . . 5
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏)) |
| 16 | 15 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHom 𝑏)) |
| 17 | | eqidd 2738 |
. . . . . 6
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) |
| 18 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏)) |
| 19 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎)) |
| 20 | 18, 19 | oveqan12rd 7451 |
. . . . . . 7
⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m
(Base‘𝑎))) |
| 21 | 20 | adantl 481 |
. . . . . 6
⊢ (((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m
(Base‘𝑎))) |
| 22 | | simpl 482 |
. . . . . 6
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → 𝑎 ∈ 𝑅) |
| 23 | | simpr 484 |
. . . . . 6
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑅) |
| 24 | | ovexd 7466 |
. . . . . 6
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V) |
| 25 | 17, 21, 22, 23, 24 | ovmpod 7585 |
. . . . 5
⊢ ((𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅) → (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎))) |
| 26 | 25 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑m (Base‘𝑎))) |
| 27 | 14, 16, 26 | 3sstr4d 4039 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑅 ∧ 𝑏 ∈ 𝑅)) → (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏)) |
| 28 | 27 | ralrimivva 3202 |
. 2
⊢ (𝜑 → ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏)) |
| 29 | | rnghmfn 20439 |
. . . . 5
⊢ RngHom
Fn (Rng × Rng) |
| 30 | 29 | a1i 11 |
. . . 4
⊢ (𝜑 → RngHom Fn (Rng ×
Rng)) |
| 31 | | rnghmsscmap.r |
. . . . . 6
⊢ (𝜑 → 𝑅 = (Rng ∩ 𝑈)) |
| 32 | | inss1 4237 |
. . . . . 6
⊢ (Rng
∩ 𝑈) ⊆
Rng |
| 33 | 31, 32 | eqsstrdi 4028 |
. . . . 5
⊢ (𝜑 → 𝑅 ⊆ Rng) |
| 34 | | xpss12 5700 |
. . . . 5
⊢ ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng ×
Rng)) |
| 35 | 33, 33, 34 | syl2anc 584 |
. . . 4
⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Rng ×
Rng)) |
| 36 | | fnssres 6691 |
. . . 4
⊢ (( RngHom
Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHom
↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅)) |
| 37 | 30, 35, 36 | syl2anc 584 |
. . 3
⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅)) |
| 38 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) |
| 39 | | ovex 7464 |
. . . . 5
⊢
((Base‘𝑦)
↑m (Base‘𝑥)) ∈ V |
| 40 | 38, 39 | fnmpoi 8095 |
. . . 4
⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑅 × 𝑅) |
| 41 | 40 | a1i 11 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) Fn (𝑅 × 𝑅)) |
| 42 | | incom 4209 |
. . . . 5
⊢ (Rng
∩ 𝑈) = (𝑈 ∩ Rng) |
| 43 | | rnghmsscmap.u |
. . . . . 6
⊢ (𝜑 → 𝑈 ∈ 𝑉) |
| 44 | | inex1g 5319 |
. . . . . 6
⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V) |
| 45 | 43, 44 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V) |
| 46 | 42, 45 | eqeltrid 2845 |
. . . 4
⊢ (𝜑 → (Rng ∩ 𝑈) ∈ V) |
| 47 | 31, 46 | eqeltrd 2841 |
. . 3
⊢ (𝜑 → 𝑅 ∈ V) |
| 48 | 37, 41, 47 | isssc 17864 |
. 2
⊢ (𝜑 → (( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ↔ (𝑅 ⊆ 𝑅 ∧ ∀𝑎 ∈ 𝑅 ∀𝑏 ∈ 𝑅 (𝑎( RngHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))𝑏)))) |
| 49 | 1, 28, 48 | mpbir2and 713 |
1
⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) |