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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opelidres | Structured version Visualization version GIF version |
Description: Characterization of the ordered pairs in the restricted identity relation when the intersection of their component belongs to the restricting class. TODO: prove bj-idreseq 37077 from it. (Contributed by BJ, 29-Mar-2020.) |
Ref | Expression |
---|---|
bj-opelidres | ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-idres 37075 | . . 3 ⊢ ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉)) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ( I ↾ 𝑉) ↔ 〈𝐴, 𝐵〉 ∈ ( I ∩ (𝑉 × 𝑉))) |
3 | elin 3986 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (〈𝐴, 𝐵〉 ∈ I ∧ 〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑉))) | |
4 | inex1g 5340 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
5 | bj-opelid 37071 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (〈𝐴, 𝐵〉 ∈ I ↔ 𝐴 = 𝐵)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ I ↔ 𝐴 = 𝐵)) |
7 | opelxp 5735 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑉) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑉) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) |
9 | 6, 8 | anbi12d 631 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((〈𝐴, 𝐵〉 ∈ I ∧ 〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑉)) ↔ (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))) |
10 | simpl 482 | . . . . 5 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 = 𝐵) | |
11 | eleq1 2826 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉)) | |
12 | 11 | biimpcd 249 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐵 ∈ 𝑉)) |
13 | 12 | anc2li 555 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) |
14 | 13 | ancld 550 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))) |
15 | 10, 14 | impbid2 226 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ↔ 𝐴 = 𝐵)) |
16 | 9, 15 | bitrd 279 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((〈𝐴, 𝐵〉 ∈ I ∧ 〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵)) |
17 | 3, 16 | bitrid 283 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ ( I ∩ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵)) |
18 | 2, 17 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 Vcvv 3482 ∩ cin 3969 〈cop 4654 I cid 5596 × cxp 5697 ↾ cres 5701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-res 5711 |
This theorem is referenced by: (None) |
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