Theorem bj-eldiag2 37617

Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 37611. (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 37615	. . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
2	1	eleq2d 2842	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴))))
3		elin 3915	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)))
4		bj-opelidb1 37593	. . . 4 ⊢ (⟨𝐵, 𝐶⟩ ∈ I ↔ (𝐵 ∈ V ∧ 𝐵 = 𝐶))
5		opelxp 5676	. . . 4 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))
6	4, 5	anbi12i 636	. . 3 ⊢ ((⟨𝐵, 𝐶⟩ ∈ I ∧ ⟨𝐵, 𝐶⟩ ∈ (𝐴 × 𝐴)) ↔ ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)))
7		simprl 778	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
8		simplr 776	. . . . 5 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 = 𝐶)
9	7, 8	jca 518	. . . 4 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))
10		elex 3469	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
11	10	anim1i 623	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ V ∧ 𝐵 = 𝐶))
12		eleq1 2844	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
13	12	biimpcd 251	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴))
14	13	imdistani 575	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴))
15	11, 14	jca 518	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶) → ((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)))
16	9, 15	impbii 211	. . 3 ⊢ (((𝐵 ∈ V ∧ 𝐵 = 𝐶) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))
17	3, 6, 16	3bitri 299	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))
18	2, 17	bitrdi 289	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1554   ∈ wcel 2136  Vcvv 3448   ∩ cin 3898  ⟨cop 4582   I cid 5534   × cxp 5638  ‘cfv 6510  Idcdiag2 37612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-res 5652  df-iota 6466  df-fun 6512  df-fv 6518  df-bj-diag 37613

This theorem is referenced by: (None)