Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-idres | Structured version Visualization version GIF version |
Description: Alternate expression for
the restricted identity relation. The
advantage of that expression is to expose it as a "bounded"
class, being
included in the Cartesian square of the restricting class. (Contributed
by BJ, 27-Dec-2023.)
This is an alternate of idinxpresid 5882 (see idinxpres 5881). See also elrid 5880 and elidinxp 5878. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-idres | ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5531 | . 2 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V)) | |
2 | inss1 4155 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ I | |
3 | relinxp 5651 | . . . . 5 ⊢ Rel ( I ∩ (𝐴 × V)) | |
4 | elin 3897 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ∩ (𝐴 × V)) ↔ (〈𝑥, 𝑦〉 ∈ I ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × V))) | |
5 | bj-opelidb1 34568 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦)) | |
6 | 5 | simprbi 500 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ I → 𝑥 = 𝑦) |
7 | opelxp1 5560 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × V) → 𝑥 ∈ 𝐴) | |
8 | simpr 488 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
9 | eleq1w 2872 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
10 | 9 | biimpa 480 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) |
11 | 8, 10 | jca 515 | . . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
12 | 6, 7, 11 | syl2an 598 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 ∈ I ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
13 | 4, 12 | sylbi 220 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ∩ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
14 | opelxpi 5556 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐴)) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ∩ (𝐴 × V)) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐴)) |
16 | 3, 15 | relssi 5624 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴) |
17 | 2, 16 | ssini 4158 | . . 3 ⊢ ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴)) |
18 | ssv 3939 | . . . 4 ⊢ 𝐴 ⊆ V | |
19 | xpss2 5539 | . . . 4 ⊢ (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V)) | |
20 | sslin 4161 | . . . 4 ⊢ ((𝐴 × 𝐴) ⊆ (𝐴 × V) → ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V))) | |
21 | 18, 19, 20 | mp2b 10 | . . 3 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V)) |
22 | 17, 21 | eqssi 3931 | . 2 ⊢ ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴)) |
23 | 1, 22 | eqtri 2821 | 1 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 〈cop 4531 I cid 5424 × cxp 5517 ↾ cres 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-res 5531 |
This theorem is referenced by: bj-opelidres 34576 |
Copyright terms: Public domain | W3C validator |