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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eldiag2 | Structured version Visualization version GIF version | ||
| Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 37172. (Contributed by BJ, 22-Jun-2019.) |
| Ref | Expression |
|---|---|
| bj-eldiag2 | ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-diagval2 37176 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴))) | |
| 2 | 1 | eleq2d 2827 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ 〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)))) |
| 3 | elin 3967 | . . 3 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (〈𝐵, 𝐶〉 ∈ I ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴))) | |
| 4 | bj-opelidb1 37154 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶)) | |
| 5 | opelxp 5721 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) | |
| 6 | 4, 5 | anbi12i 628 | . . 3 ⊢ ((〈𝐵, 𝐶〉 ∈ I ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))) |
| 7 | simprl 771 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴) | |
| 8 | simplr 769 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 = 𝐶) | |
| 9 | 7, 8 | jca 511 | . . . 4 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
| 10 | elex 3501 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
| 11 | 10 | anim1i 615 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶)) |
| 12 | eleq1 2829 | . . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
| 13 | 12 | biimpcd 249 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴)) |
| 14 | 13 | imdistani 568 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) |
| 15 | 11, 14 | jca 511 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))) |
| 16 | 9, 15 | impbii 209 | . . 3 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
| 17 | 3, 6, 16 | 3bitri 297 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
| 18 | 2, 17 | bitrdi 287 | 1 ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 〈cop 4632 I cid 5577 × cxp 5683 ‘cfv 6561 Idcdiag2 37173 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-bj-diag 37174 |
| This theorem is referenced by: (None) |
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