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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. (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
bj-eldiag2 | ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-diagval 33090 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴))) | |
2 | 1 | eleq2d 2687 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Diag‘𝐴) ↔ 〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)))) |
3 | elin 3796 | . . 3 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (〈𝐵, 𝐶〉 ∈ I ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴))) | |
4 | bj-elid3 33087 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶)) | |
5 | opelxp 5146 | . . . 4 ⊢ (〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) | |
6 | 4, 5 | anbi12i 733 | . . 3 ⊢ ((〈𝐵, 𝐶〉 ∈ I ∧ 〈𝐵, 𝐶〉 ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))) |
7 | simprl 794 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴) | |
8 | simplr 792 | . . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 = 𝐶) | |
9 | 7, 8 | jca 554 | . . . 4 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
10 | elex 3212 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V) | |
11 | 10 | anim1i 592 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶)) |
12 | eleq1 2689 | . . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
13 | 12 | biimpcd 239 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴)) |
14 | 13 | imdistani 726 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) |
15 | 11, 14 | jca 554 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))) |
16 | 9, 15 | impbii 199 | . . 3 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
17 | 3, 6, 16 | 3bitri 286 | . 2 ⊢ (〈𝐵, 𝐶〉 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)) |
18 | 2, 17 | syl6bb 276 | 1 ⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 〈cop 4183 I cid 5023 × cxp 5112 ‘cfv 5888 Diagcdiag2 33088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-2nd 7169 df-bj-diag 33089 |
This theorem is referenced by: (None) |
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