Theorem bj-eldiag2 36053

Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 36047. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-eldiag2	⊢ (𝐴 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)))

Proof of Theorem bj-eldiag2

Step	Hyp	Ref	Expression
1		bj-diagval2 36051	. . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
2	1	eleq2d 2819	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴))))
3		elin 3964	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)))
4		bj-opelidb1 36029	. . . 4 ⊢ (⟨𝐵, 𝐶⟩ ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶))
5		opelxp 5712	. . . 4 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))
6	4, 5	anbi12i 627	. . 3 ⊢ ((⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)))
7		simprl 769	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
8		simplr 767	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 = 𝐶)
9	7, 8	jca 512	. . . 4 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))
10		elex 3492	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
11	10	anim1i 615	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶))
12		eleq1 2821	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
13	12	biimpcd 248	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴))
14	13	imdistani 569	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))
15	11, 14	jca 512	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)))
16	9, 15	impbii 208	. . 3 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))
17	3, 6, 16	3bitri 296	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))
18	2, 17	bitrdi 286	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3947  ⟨cop 4634   I cid 5573   × cxp 5674  ‘cfv 6543  Idcdiag2 36048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-bj-diag 36049

This theorem is referenced by: (None)