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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-eldiag | Structured version Visualization version GIF version |
Description: Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
bj-eldiag | ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-diagval 32267 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴))) | |
2 | 1 | eleq2d 2673 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ 𝐵 ∈ ( I ∩ (𝐴 × 𝐴)))) |
3 | elin 3758 | . . 3 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴))) | |
4 | ancom 465 | . . 3 ⊢ ((𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I )) | |
5 | bj-elid2 32263 | . . . 4 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 ∈ I ↔ (1st ‘𝐵) = (2nd ‘𝐵))) | |
6 | 5 | pm5.32i 667 | . . 3 ⊢ ((𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))) |
7 | 3, 4, 6 | 3bitri 285 | . 2 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))) |
8 | 2, 7 | syl6bb 275 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 I cid 4948 × cxp 5036 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 Diagcdiag2 32265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 df-bj-diag 32266 |
This theorem is referenced by: (None) |
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