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Mirrors > Home > ILE Home > Th. List > mptresid | GIF version |
Description: The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.) |
Ref | Expression |
---|---|
mptresid | ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 4029 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
2 | opabresid 4921 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴) | |
3 | 1, 2 | eqtri 2178 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 ∈ wcel 2128 {copab 4026 ↦ cmpt 4027 I cid 4250 ↾ cres 4590 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-res 4600 |
This theorem is referenced by: idref 5709 restid2 12430 txswaphmeolem 12790 dvexp 13145 dvmptidcn 13148 |
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