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Mirrors > Home > ILE Home > Th. List > ecinxp | GIF version |
Description: Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.) |
Ref | Expression |
---|---|
ecinxp | ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . . . . 8 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | |
2 | 1 | snssd 3635 | . . . . . . 7 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴) |
3 | df-ss 3054 | . . . . . . 7 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵}) | |
4 | 2, 3 | sylib 121 | . . . . . 6 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → ({𝐵} ∩ 𝐴) = {𝐵}) |
5 | 4 | imaeq2d 4851 | . . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ ({𝐵} ∩ 𝐴)) = (𝑅 “ {𝐵})) |
6 | 5 | ineq1d 3246 | . . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴) = ((𝑅 “ {𝐵}) ∩ 𝐴)) |
7 | imass2 4885 | . . . . . . 7 ⊢ ({𝐵} ⊆ 𝐴 → (𝑅 “ {𝐵}) ⊆ (𝑅 “ 𝐴)) | |
8 | 2, 7 | syl 14 | . . . . . 6 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ {𝐵}) ⊆ (𝑅 “ 𝐴)) |
9 | simpl 108 | . . . . . 6 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ 𝐴) ⊆ 𝐴) | |
10 | 8, 9 | sstrd 3077 | . . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ {𝐵}) ⊆ 𝐴) |
11 | df-ss 3054 | . . . . 5 ⊢ ((𝑅 “ {𝐵}) ⊆ 𝐴 ↔ ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵})) | |
12 | 10, 11 | sylib 121 | . . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝑅 “ {𝐵}) ∩ 𝐴) = (𝑅 “ {𝐵})) |
13 | 6, 12 | eqtr2d 2151 | . . 3 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴)) |
14 | imainrect 4954 | . . 3 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}) = ((𝑅 “ ({𝐵} ∩ 𝐴)) ∩ 𝐴) | |
15 | 13, 14 | syl6eqr 2168 | . 2 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑅 “ {𝐵}) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵})) |
16 | df-ec 6399 | . 2 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
17 | df-ec 6399 | . 2 ⊢ [𝐵](𝑅 ∩ (𝐴 × 𝐴)) = ((𝑅 ∩ (𝐴 × 𝐴)) “ {𝐵}) | |
18 | 15, 16, 17 | 3eqtr4g 2175 | 1 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ∩ cin 3040 ⊆ wss 3041 {csn 3497 × cxp 4507 “ cima 4512 [cec 6395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-ec 6399 |
This theorem is referenced by: qsinxp 6473 nqnq0pi 7214 |
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