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Theorem bj-eldiag 33428
 Description: Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-eldiag (𝐴𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵))))

Proof of Theorem bj-eldiag
StepHypRef Expression
1 bj-diagval 33427 . . 3 (𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
21eleq2d 2836 . 2 (𝐴𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ 𝐵 ∈ ( I ∩ (𝐴 × 𝐴))))
3 elin 3947 . . 3 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)))
4 ancom 452 . . 3 ((𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ))
5 bj-elid2 33423 . . . 4 (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 ∈ I ↔ (1st𝐵) = (2nd𝐵)))
65pm5.32i 564 . . 3 ((𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵)))
73, 4, 63bitri 286 . 2 (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵)))
82, 7syl6bb 276 1 (𝐴𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ∩ cin 3722   I cid 5156   × cxp 5247  ‘cfv 6031  1st c1st 7313  2nd c2nd 7314  Diagcdiag2 33425 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7315  df-2nd 7316  df-bj-diag 33426 This theorem is referenced by: (None)
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