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Mirrors > Home > ILE Home > Th. List > ressn | GIF version |
Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
ressn | ⊢ (A ↾ {B}) = ({B} × (A “ {B})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4582 | . 2 ⊢ Rel (A ↾ {B}) | |
2 | relxp 4390 | . 2 ⊢ Rel ({B} × (A “ {B})) | |
3 | ancom 253 | . . . 4 ⊢ ((〈x, y〉 ∈ A ∧ x ∈ {B}) ↔ (x ∈ {B} ∧ 〈x, y〉 ∈ A)) | |
4 | vex 2554 | . . . . . . 7 ⊢ x ∈ V | |
5 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
6 | 4, 5 | elimasn 4635 | . . . . . 6 ⊢ (y ∈ (A “ {x}) ↔ 〈x, y〉 ∈ A) |
7 | elsni 3391 | . . . . . . . . 9 ⊢ (x ∈ {B} → x = B) | |
8 | 7 | sneqd 3380 | . . . . . . . 8 ⊢ (x ∈ {B} → {x} = {B}) |
9 | 8 | imaeq2d 4611 | . . . . . . 7 ⊢ (x ∈ {B} → (A “ {x}) = (A “ {B})) |
10 | 9 | eleq2d 2104 | . . . . . 6 ⊢ (x ∈ {B} → (y ∈ (A “ {x}) ↔ y ∈ (A “ {B}))) |
11 | 6, 10 | syl5bbr 183 | . . . . 5 ⊢ (x ∈ {B} → (〈x, y〉 ∈ A ↔ y ∈ (A “ {B}))) |
12 | 11 | pm5.32i 427 | . . . 4 ⊢ ((x ∈ {B} ∧ 〈x, y〉 ∈ A) ↔ (x ∈ {B} ∧ y ∈ (A “ {B}))) |
13 | 3, 12 | bitri 173 | . . 3 ⊢ ((〈x, y〉 ∈ A ∧ x ∈ {B}) ↔ (x ∈ {B} ∧ y ∈ (A “ {B}))) |
14 | 5 | opelres 4560 | . . 3 ⊢ (〈x, y〉 ∈ (A ↾ {B}) ↔ (〈x, y〉 ∈ A ∧ x ∈ {B})) |
15 | opelxp 4317 | . . 3 ⊢ (〈x, y〉 ∈ ({B} × (A “ {B})) ↔ (x ∈ {B} ∧ y ∈ (A “ {B}))) | |
16 | 13, 14, 15 | 3bitr4i 201 | . 2 ⊢ (〈x, y〉 ∈ (A ↾ {B}) ↔ 〈x, y〉 ∈ ({B} × (A “ {B}))) |
17 | 1, 2, 16 | eqrelriiv 4377 | 1 ⊢ (A ↾ {B}) = ({B} × (A “ {B})) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∈ wcel 1390 {csn 3367 〈cop 3370 × cxp 4286 ↾ cres 4290 “ cima 4291 |
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-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 |
This theorem is referenced by: (None) |
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