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Theorem bj-eldiag2 32269
 Description: Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-eldiag2 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))

Proof of Theorem bj-eldiag2
StepHypRef Expression
1 bj-diagval 32267 . . 3 (𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
21eleq2d 2673 . 2 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴))))
3 elin 3758 . . 3 (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)))
4 bj-elid3 32264 . . . 4 (⟨𝐵, 𝐶⟩ ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶))
5 opelxp 5070 . . . 4 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) ↔ (𝐵𝐴𝐶𝐴))
64, 5anbi12i 729 . . 3 ((⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)))
7 simprl 790 . . . . 5 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → 𝐵𝐴)
8 simplr 788 . . . . 5 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 = 𝐶)
97, 8jca 553 . . . 4 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐴𝐵 = 𝐶))
10 elex 3185 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
1110anim1i 590 . . . . 5 ((𝐵𝐴𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶))
12 eleq1 2676 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝐴𝐶𝐴))
1312biimpcd 238 . . . . . 6 (𝐵𝐴 → (𝐵 = 𝐶𝐶𝐴))
1413imdistani 722 . . . . 5 ((𝐵𝐴𝐵 = 𝐶) → (𝐵𝐴𝐶𝐴))
1511, 14jca 553 . . . 4 ((𝐵𝐴𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)))
169, 15impbii 198 . . 3 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) ↔ (𝐵𝐴𝐵 = 𝐶))
173, 6, 163bitri 285 . 2 (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵𝐴𝐵 = 𝐶))
182, 17syl6bb 275 1 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ⟨cop 4131   I cid 4948   × cxp 5036  ‘cfv 5804  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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