Theorem bj-eldiag2 35648

Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 35642. (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 35646	. . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
2	1	eleq2d 2823	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴))))
3		elin 3926	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)))
4		bj-opelidb1 35624	. . . 4 ⊢ (⟨𝐵, 𝐶⟩ ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶))
5		opelxp 5669	. . . 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 3463	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
11	10	anim1i 615	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶))
12		eleq1 2825	. . . . . . 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 3445   ∩ cin 3909  ⟨cop 4592   I cid 5530   × cxp 5631  ‘cfv 6496  Idcdiag2 35643

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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-res 5645  df-iota 6448  df-fun 6498  df-fv 6504  df-bj-diag 35644

This theorem is referenced by: (None)