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Mirrors > Home > ILE Home > Th. List > dfrab3ss | GIF version |
Description: Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.) |
Ref | Expression |
---|---|
dfrab3ss | ⊢ (A ⊆ B → {x ∈ A ∣ φ} = (A ∩ {x ∈ B ∣ φ})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 2925 | . . 3 ⊢ (A ⊆ B ↔ (A ∩ B) = A) | |
2 | ineq1 3125 | . . . 4 ⊢ ((A ∩ B) = A → ((A ∩ B) ∩ {x ∣ φ}) = (A ∩ {x ∣ φ})) | |
3 | 2 | eqcomd 2042 | . . 3 ⊢ ((A ∩ B) = A → (A ∩ {x ∣ φ}) = ((A ∩ B) ∩ {x ∣ φ})) |
4 | 1, 3 | sylbi 114 | . 2 ⊢ (A ⊆ B → (A ∩ {x ∣ φ}) = ((A ∩ B) ∩ {x ∣ φ})) |
5 | dfrab3 3207 | . 2 ⊢ {x ∈ A ∣ φ} = (A ∩ {x ∣ φ}) | |
6 | dfrab3 3207 | . . . 4 ⊢ {x ∈ B ∣ φ} = (B ∩ {x ∣ φ}) | |
7 | 6 | ineq2i 3129 | . . 3 ⊢ (A ∩ {x ∈ B ∣ φ}) = (A ∩ (B ∩ {x ∣ φ})) |
8 | inass 3141 | . . 3 ⊢ ((A ∩ B) ∩ {x ∣ φ}) = (A ∩ (B ∩ {x ∣ φ})) | |
9 | 7, 8 | eqtr4i 2060 | . 2 ⊢ (A ∩ {x ∈ B ∣ φ}) = ((A ∩ B) ∩ {x ∣ φ}) |
10 | 4, 5, 9 | 3eqtr4g 2094 | 1 ⊢ (A ⊆ B → {x ∈ A ∣ φ} = (A ∩ {x ∈ B ∣ φ})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 {cab 2023 {crab 2304 ∩ cin 2910 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 df-v 2553 df-in 2918 df-ss 2925 |
This theorem is referenced by: (None) |
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