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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 34457 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 34455 | . . 3 ⊢ ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉)) | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ( I ↾ 𝑉) ↔ 〈𝐴, 𝐵〉 ∈ ( I ∩ (𝑉 × 𝑉))) |
3 | elin 4169 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (〈𝐴, 𝐵〉 ∈ I ∧ 〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑉))) | |
4 | inex1g 5223 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
5 | bj-opelid 34451 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (〈𝐴, 𝐵〉 ∈ I ↔ 𝐴 = 𝐵)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ I ↔ 𝐴 = 𝐵)) |
7 | opelxp 5591 | . . . . . 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 632 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((〈𝐴, 𝐵〉 ∈ I ∧ 〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑉)) ↔ (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))) |
10 | simpl 485 | . . . . 5 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 = 𝐵) | |
11 | eleq1 2900 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉)) | |
12 | 11 | biimpcd 251 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐵 ∈ 𝑉)) |
13 | 12 | anc2li 558 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) |
14 | 13 | ancld 553 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))) |
15 | 10, 14 | impbid2 228 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ↔ 𝐴 = 𝐵)) |
16 | 9, 15 | bitrd 281 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((〈𝐴, 𝐵〉 ∈ I ∧ 〈𝐴, 𝐵〉 ∈ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵)) |
17 | 3, 16 | syl5bb 285 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ ( I ∩ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵)) |
18 | 2, 17 | syl5bb 285 | 1 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 〈cop 4573 I cid 5459 × cxp 5553 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-res 5567 |
This theorem is referenced by: (None) |
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