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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 37114. (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 37118 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴))) | |
2 | 1 | eleq2d 2823 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ 〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)))) |
3 | elin 3979 | . . 3 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (〈𝐵, 𝐶〉 ∈ I ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴))) | |
4 | bj-opelidb1 37096 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶)) | |
5 | opelxp 5720 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) | |
6 | 4, 5 | anbi12i 627 | . . 3 ⊢ ((〈𝐵, 𝐶〉 ∈ I ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))) |
7 | simprl 770 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴) | |
8 | simplr 768 | . . . . 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 3498 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
11 | 10 | anim1i 614 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶)) |
12 | eleq1 2825 | . . . . . . 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 1535 ∈ wcel 2104 Vcvv 3477 ∩ cin 3962 〈cop 4637 I cid 5576 × cxp 5682 ‘cfv 6559 Idcdiag2 37115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7748 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4916 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-iota 6511 df-fun 6561 df-fv 6567 df-bj-diag 37116 |
This theorem is referenced by: (None) |
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