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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 34355. (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 34359 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴))) | |
2 | 1 | eleq2d 2895 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ 〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)))) |
3 | elin 4166 | . . 3 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (〈𝐵, 𝐶〉 ∈ I ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴))) | |
4 | bj-opelidb1 34337 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶)) | |
5 | opelxp 5584 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) | |
6 | 4, 5 | anbi12i 626 | . . 3 ⊢ ((〈𝐵, 𝐶〉 ∈ I ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))) |
7 | simprl 767 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴) | |
8 | simplr 765 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 = 𝐶) | |
9 | 7, 8 | jca 512 | . . . 4 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
10 | elex 3510 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
11 | 10 | anim1i 614 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶)) |
12 | eleq1 2897 | . . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
13 | 12 | biimpcd 250 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴)) |
14 | 13 | imdistani 569 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) |
15 | 11, 14 | jca 512 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))) |
16 | 9, 15 | impbii 210 | . . 3 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
17 | 3, 6, 16 | 3bitri 298 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
18 | 2, 17 | syl6bb 288 | 1 ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 〈cop 4563 I cid 5452 × cxp 5546 ‘cfv 6348 Idcdiag2 34356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-bj-diag 34357 |
This theorem is referenced by: (None) |
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