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Theorem bj-eldiag2 33422
Description: Characterization of the elements of 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 33420 . . 3 (𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
21eleq2d 2835 . 2 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴))))
3 elin 3945 . . 3 (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)))
4 bj-elid3 33417 . . . 4 (⟨𝐵, 𝐶⟩ ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶))
5 opelxp 5286 . . . 4 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) ↔ (𝐵𝐴𝐶𝐴))
64, 5anbi12i 604 . . 3 ((⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)))
7 simprl 746 . . . . 5 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → 𝐵𝐴)
8 simplr 744 . . . . 5 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 = 𝐶)
97, 8jca 495 . . . 4 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐴𝐵 = 𝐶))
10 elex 3361 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
1110anim1i 594 . . . . 5 ((𝐵𝐴𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶))
12 eleq1 2837 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝐴𝐶𝐴))
1312biimpcd 239 . . . . . 6 (𝐵𝐴 → (𝐵 = 𝐶𝐶𝐴))
1413imdistani 550 . . . . 5 ((𝐵𝐴𝐵 = 𝐶) → (𝐵𝐴𝐶𝐴))
1511, 14jca 495 . . . 4 ((𝐵𝐴𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)))
169, 15impbii 199 . . 3 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) ↔ (𝐵𝐴𝐵 = 𝐶))
173, 6, 163bitri 286 . 2 (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵𝐴𝐵 = 𝐶))
182, 17syl6bb 276 1 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  cin 3720  cop 4320   I cid 5156   × cxp 5247  cfv 6031  Diagcdiag2 33418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  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 7314  df-2nd 7315  df-bj-diag 33419
This theorem is referenced by: (None)
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