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Theorem bj-eldiag2 36565
Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 36559. (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 36563 . . 3 (𝐴𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
21eleq2d 2813 . 2 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴))))
3 elin 3959 . . 3 (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)))
4 bj-opelidb1 36541 . . . 4 (⟨𝐵, 𝐶⟩ ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶))
5 opelxp 5705 . . . 4 (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) ↔ (𝐵𝐴𝐶𝐴))
64, 5anbi12i 626 . . 3 ((⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)))
7 simprl 768 . . . . 5 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → 𝐵𝐴)
8 simplr 766 . . . . 5 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 = 𝐶)
97, 8jca 511 . . . 4 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐴𝐵 = 𝐶))
10 elex 3487 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
1110anim1i 614 . . . . 5 ((𝐵𝐴𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶))
12 eleq1 2815 . . . . . . 7 (𝐵 = 𝐶 → (𝐵𝐴𝐶𝐴))
1312biimpcd 248 . . . . . 6 (𝐵𝐴 → (𝐵 = 𝐶𝐶𝐴))
1413imdistani 568 . . . . 5 ((𝐵𝐴𝐵 = 𝐶) → (𝐵𝐴𝐶𝐴))
1511, 14jca 511 . . . 4 ((𝐵𝐴𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)))
169, 15impbii 208 . . 3 (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵𝐴𝐶𝐴)) ↔ (𝐵𝐴𝐵 = 𝐶))
173, 6, 163bitri 297 . 2 (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵𝐴𝐵 = 𝐶))
182, 17bitrdi 287 1 (𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  cin 3942  cop 4629   I cid 5566   × cxp 5667  cfv 6537  Idcdiag2 36560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6489  df-fun 6539  df-fv 6545  df-bj-diag 36561
This theorem is referenced by: (None)
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