Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
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 34457. (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 34461 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴))) | |
2 | 1 | eleq2d 2898 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ 〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)))) |
3 | elin 4169 | . . 3 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (〈𝐵, 𝐶〉 ∈ I ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴))) | |
4 | bj-opelidb1 34439 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶)) | |
5 | opelxp 5586 | . . . 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 769 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴) | |
8 | simplr 767 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 = 𝐶) | |
9 | 7, 8 | jca 514 | . . . 4 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
10 | elex 3513 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
11 | 10 | anim1i 616 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶)) |
12 | eleq1 2900 | . . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
13 | 12 | biimpcd 251 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴)) |
14 | 13 | imdistani 571 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) |
15 | 11, 14 | jca 514 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))) |
16 | 9, 15 | impbii 211 | . . 3 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
17 | 3, 6, 16 | 3bitri 299 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
18 | 2, 17 | syl6bb 289 | 1 ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∩ cin 3935 〈cop 4567 I cid 5454 × cxp 5548 ‘cfv 6350 Idcdiag2 34458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-res 5562 df-iota 6309 df-fun 6352 df-fv 6358 df-bj-diag 34459 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |