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Theorem bj-eldiag2 34361
Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 34355. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-eldiag2 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))

Proof of Theorem bj-eldiag2
StepHypRef Expression
1 bj-diagval2 34359 . . 3 (𝐴𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
21eleq2d 2895 . 2 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴))))
3 elin 4166 . . 3 (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)))
4 bj-opelidb1 34337 . . . 4 (⟨𝐵, 𝐶⟩ ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶))
5 opelxp 5584 . . . 4 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) ↔ (𝐵𝐴𝐶𝐴))
64, 5anbi12i 626 . . 3 ((⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)))
7 simprl 767 . . . . 5 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → 𝐵𝐴)
8 simplr 765 . . . . 5 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 = 𝐶)
97, 8jca 512 . . . 4 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐴𝐵 = 𝐶))
10 elex 3510 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
1110anim1i 614 . . . . 5 ((𝐵𝐴𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶))
12 eleq1 2897 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝐴𝐶𝐴))
1312biimpcd 250 . . . . . 6 (𝐵𝐴 → (𝐵 = 𝐶𝐶𝐴))
1413imdistani 569 . . . . 5 ((𝐵𝐴𝐵 = 𝐶) → (𝐵𝐴𝐶𝐴))
1511, 14jca 512 . . . 4 ((𝐵𝐴𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)))
169, 15impbii 210 . . 3 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) ↔ (𝐵𝐴𝐵 = 𝐶))
173, 6, 163bitri 298 . 2 (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵𝐴𝐵 = 𝐶))
182, 17syl6bb 288 1 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  cop 4563   I cid 5452   × cxp 5546  cfv 6348  Idcdiag2 34356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-bj-diag 34357
This theorem is referenced by: (None)
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