Step | Hyp | Ref
| Expression |
1 | | off2.2 |
. . . . . 6
⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
2 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝑆) |
3 | | off2.6 |
. . . . . . 7
⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶) |
4 | | inss1 3795 |
. . . . . . 7
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 |
5 | 3, 4 | syl6eqssr 3619 |
. . . . . 6
⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
6 | 5 | sselda 3568 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴) |
7 | 2, 6 | ffvelrnd 6268 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
8 | | off2.3 |
. . . . . 6
⊢ (𝜑 → 𝐺:𝐵⟶𝑇) |
9 | 8 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐺:𝐵⟶𝑇) |
10 | | inss2 3796 |
. . . . . . 7
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 |
11 | 3, 10 | syl6eqssr 3619 |
. . . . . 6
⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
12 | 11 | sselda 3568 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵) |
13 | 9, 12 | ffvelrnd 6268 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇) |
14 | | off2.1 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) |
15 | 14 | ralrimivva 2954 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
16 | 15 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
17 | | ovrspc2v 6571 |
. . . 4
⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ (𝐺‘𝑧) ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
18 | 7, 13, 16, 17 | syl21anc 1317 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
19 | | eqid 2610 |
. . 3
⊢ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) |
20 | 18, 19 | fmptd 6292 |
. 2
⊢ (𝜑 → (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈) |
21 | | ffn 5958 |
. . . . . 6
⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴) |
22 | 1, 21 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐹 Fn 𝐴) |
23 | | ffn 5958 |
. . . . . 6
⊢ (𝐺:𝐵⟶𝑇 → 𝐺 Fn 𝐵) |
24 | 8, 23 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐺 Fn 𝐵) |
25 | | off2.4 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
26 | | off2.5 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ 𝑊) |
27 | | eqid 2610 |
. . . . 5
⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵) |
28 | | eqidd 2611 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) |
29 | | eqidd 2611 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) |
30 | 22, 24, 25, 26, 27, 28, 29 | offval 6802 |
. . . 4
⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
31 | 3 | mpteq1d 4666 |
. . . 4
⊢ (𝜑 → (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
32 | 30, 31 | eqtrd 2644 |
. . 3
⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
33 | 32 | feq1d 5943 |
. 2
⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈 ↔ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈)) |
34 | 20, 33 | mpbird 246 |
1
⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈) |