Proof of Theorem ressval3d
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ressval3d.u | . 2
⊢ (𝜑 → 𝐴 ⊆ 𝐵) | 
| 2 |  | sspss 4102 | . . . 4
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) | 
| 3 |  | dfpss3 4089 | . . . . 5
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) | 
| 4 | 3 | orbi1i 914 | . . . 4
⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵)) | 
| 5 | 2, 4 | bitri 275 | . . 3
⊢ (𝐴 ⊆ 𝐵 ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵)) | 
| 6 |  | simplr 769 | . . . . . . 7
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → ¬ 𝐵 ⊆ 𝐴) | 
| 7 |  | ressval3d.s | . . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ 𝑉) | 
| 8 | 7 | adantl 481 | . . . . . . 7
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑆 ∈ 𝑉) | 
| 9 |  | simpl 482 | . . . . . . . 8
⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝐵) | 
| 10 |  | ressval3d.b | . . . . . . . . . 10
⊢ 𝐵 = (Base‘𝑆) | 
| 11 | 10 | fvexi 6920 | . . . . . . . . 9
⊢ 𝐵 ∈ V | 
| 12 | 11 | a1i 11 | . . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ V) | 
| 13 |  | ssexg 5323 | . . . . . . . 8
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | 
| 14 | 9, 12, 13 | syl2an 596 | . . . . . . 7
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 ∈ V) | 
| 15 |  | ressval3d.r | . . . . . . . 8
⊢ 𝑅 = (𝑆 ↾s 𝐴) | 
| 16 | 15, 10 | ressval2 17279 | . . . . . . 7
⊢ ((¬
𝐵 ⊆ 𝐴 ∧ 𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) → 𝑅 = (𝑆 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) | 
| 17 | 6, 8, 14, 16 | syl3anc 1373 | . . . . . 6
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉)) | 
| 18 |  | ressval3d.e | . . . . . . . . . 10
⊢ 𝐸 =
(Base‘ndx) | 
| 19 | 18 | a1i 11 | . . . . . . . . 9
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐸 = (Base‘ndx)) | 
| 20 |  | dfss2 3969 | . . . . . . . . . . . . 13
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | 
| 21 | 20 | biimpi 216 | . . . . . . . . . . . 12
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) | 
| 22 | 21 | eqcomd 2743 | . . . . . . . . . . 11
⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐴 ∩ 𝐵)) | 
| 23 | 22 | adantr 480 | . . . . . . . . . 10
⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐴 ∩ 𝐵)) | 
| 24 | 23 | adantr 480 | . . . . . . . . 9
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝐴 = (𝐴 ∩ 𝐵)) | 
| 25 | 19, 24 | opeq12d 4881 | . . . . . . . 8
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 〈𝐸, 𝐴〉 = 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉) | 
| 26 | 25 | eqcomd 2743 | . . . . . . 7
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉 = 〈𝐸, 𝐴〉) | 
| 27 | 26 | oveq2d 7447 | . . . . . 6
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → (𝑆 sSet 〈(Base‘ndx), (𝐴 ∩ 𝐵)〉) = (𝑆 sSet 〈𝐸, 𝐴〉)) | 
| 28 | 17, 27 | eqtrd 2777 | . . . . 5
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∧ 𝜑) → 𝑅 = (𝑆 sSet 〈𝐸, 𝐴〉)) | 
| 29 | 28 | ex 412 | . . . 4
⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) → (𝜑 → 𝑅 = (𝑆 sSet 〈𝐸, 𝐴〉))) | 
| 30 | 15 | a1i 11 | . . . . . . 7
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = (𝑆 ↾s 𝐴)) | 
| 31 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝐴 = 𝐵 → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵)) | 
| 32 | 31 | adantr 480 | . . . . . . 7
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 ↾s 𝐴) = (𝑆 ↾s 𝐵)) | 
| 33 | 7 | adantl 481 | . . . . . . . 8
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 ∈ 𝑉) | 
| 34 | 10 | ressid 17290 | . . . . . . . 8
⊢ (𝑆 ∈ 𝑉 → (𝑆 ↾s 𝐵) = 𝑆) | 
| 35 | 33, 34 | syl 17 | . . . . . . 7
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 ↾s 𝐵) = 𝑆) | 
| 36 | 30, 32, 35 | 3eqtrd 2781 | . . . . . 6
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = 𝑆) | 
| 37 |  | baseid 17250 | . . . . . . . 8
⊢ Base =
Slot (Base‘ndx) | 
| 38 |  | ressval3d.f | . . . . . . . 8
⊢ (𝜑 → Fun 𝑆) | 
| 39 |  | ressval3d.d | . . . . . . . . 9
⊢ (𝜑 → 𝐸 ∈ dom 𝑆) | 
| 40 | 18, 39 | eqeltrrid 2846 | . . . . . . . 8
⊢ (𝜑 → (Base‘ndx) ∈
dom 𝑆) | 
| 41 | 37, 7, 38, 40 | setsidvald 17236 | . . . . . . 7
⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(Base‘ndx),
(Base‘𝑆)〉)) | 
| 42 | 41 | adantl 481 | . . . . . 6
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑆 = (𝑆 sSet 〈(Base‘ndx),
(Base‘𝑆)〉)) | 
| 43 | 18 | a1i 11 | . . . . . . . . 9
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐸 = (Base‘ndx)) | 
| 44 |  | simpl 482 | . . . . . . . . . 10
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = 𝐵) | 
| 45 | 44, 10 | eqtrdi 2793 | . . . . . . . . 9
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝐴 = (Base‘𝑆)) | 
| 46 | 43, 45 | opeq12d 4881 | . . . . . . . 8
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 〈𝐸, 𝐴〉 = 〈(Base‘ndx),
(Base‘𝑆)〉) | 
| 47 | 46 | eqcomd 2743 | . . . . . . 7
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 〈(Base‘ndx),
(Base‘𝑆)〉 =
〈𝐸, 𝐴〉) | 
| 48 | 47 | oveq2d 7447 | . . . . . 6
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝑆 sSet 〈(Base‘ndx),
(Base‘𝑆)〉) =
(𝑆 sSet 〈𝐸, 𝐴〉)) | 
| 49 | 36, 42, 48 | 3eqtrd 2781 | . . . . 5
⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝑅 = (𝑆 sSet 〈𝐸, 𝐴〉)) | 
| 50 | 49 | ex 412 | . . . 4
⊢ (𝐴 = 𝐵 → (𝜑 → 𝑅 = (𝑆 sSet 〈𝐸, 𝐴〉))) | 
| 51 | 29, 50 | jaoi 858 | . . 3
⊢ (((𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) ∨ 𝐴 = 𝐵) → (𝜑 → 𝑅 = (𝑆 sSet 〈𝐸, 𝐴〉))) | 
| 52 | 5, 51 | sylbi 217 | . 2
⊢ (𝐴 ⊆ 𝐵 → (𝜑 → 𝑅 = (𝑆 sSet 〈𝐸, 𝐴〉))) | 
| 53 | 1, 52 | mpcom 38 | 1
⊢ (𝜑 → 𝑅 = (𝑆 sSet 〈𝐸, 𝐴〉)) |