bj-elid6

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

		Ref	Expression
	Assertion	bj-elid6	⊢ ( 𝐵 ∈ ( I ↾ 𝐴 ) ↔ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ∧ ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-res	⊢ ( I ↾ 𝐴 ) = ( I ∩ ( 𝐴 × V ) )
2	1	elin2	⊢ ( 𝐵 ∈ ( I ↾ 𝐴 ) ↔ ( 𝐵 ∈ I ∧ 𝐵 ∈ ( 𝐴 × V ) ) )
3	2	biancomi	⊢ ( 𝐵 ∈ ( I ↾ 𝐴 ) ↔ ( 𝐵 ∈ ( 𝐴 × V ) ∧ 𝐵 ∈ I ) )
4		bj-elid4	⊢ ( 𝐵 ∈ ( 𝐴 × V ) → ( 𝐵 ∈ I ↔ ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ) )
5	4	pm5.32i	⊢ ( ( 𝐵 ∈ ( 𝐴 × V ) ∧ 𝐵 ∈ I ) ↔ ( 𝐵 ∈ ( 𝐴 × V ) ∧ ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ) )
6		1st2nd2	⊢ ( 𝐵 ∈ ( 𝐴 × V ) → 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ )
7	6	pm4.71ri	⊢ ( 𝐵 ∈ ( 𝐴 × V ) ↔ ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∧ 𝐵 ∈ ( 𝐴 × V ) ) )
8		eleq1	⊢ ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ → ( 𝐵 ∈ ( 𝐴 × V ) ↔ ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∈ ( 𝐴 × V ) ) )
9	8	adantl	⊢ ( ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ∧ 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ) → ( 𝐵 ∈ ( 𝐴 × V ) ↔ ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∈ ( 𝐴 × V ) ) )
10		simpl	⊢ ( ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ V ) → ( 1^st ‘ 𝐵 ) ∈ 𝐴 )
11	10	a1i	⊢ ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) → ( ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ V ) → ( 1^st ‘ 𝐵 ) ∈ 𝐴 ) )
12		eleq1	⊢ ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) → ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ↔ ( 2^nd ‘ 𝐵 ) ∈ 𝐴 ) )
13	10 12	syl5ib	⊢ ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) → ( ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ V ) → ( 2^nd ‘ 𝐵 ) ∈ 𝐴 ) )
14	11 13	jcad	⊢ ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) → ( ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ V ) → ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ 𝐴 ) ) )
15		elex	⊢ ( ( 2^nd ‘ 𝐵 ) ∈ 𝐴 → ( 2^nd ‘ 𝐵 ) ∈ V )
16	15	anim2i	⊢ ( ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ 𝐴 ) → ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ V ) )
17	14 16	impbid1	⊢ ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) → ( ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ V ) ↔ ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ 𝐴 ) ) )
18	17	adantr	⊢ ( ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ∧ 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ) → ( ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ V ) ↔ ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ 𝐴 ) ) )
19		opelxp	⊢ ( ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∈ ( 𝐴 × V ) ↔ ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ V ) )
20		opelxp	⊢ ( ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∈ ( 𝐴 × 𝐴 ) ↔ ( ( 1^st ‘ 𝐵 ) ∈ 𝐴 ∧ ( 2^nd ‘ 𝐵 ) ∈ 𝐴 ) )
21	18 19 20	3bitr4g	⊢ ( ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ∧ 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ) → ( ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∈ ( 𝐴 × V ) ↔ ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∈ ( 𝐴 × 𝐴 ) ) )
22		eleq1	⊢ ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ → ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ↔ ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∈ ( 𝐴 × 𝐴 ) ) )
23	22	bicomd	⊢ ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ → ( ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∈ ( 𝐴 × 𝐴 ) ↔ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) )
24	23	adantl	⊢ ( ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ∧ 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ) → ( ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∈ ( 𝐴 × 𝐴 ) ↔ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) )
25	9 21 24	3bitrd	⊢ ( ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ∧ 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ) → ( 𝐵 ∈ ( 𝐴 × V ) ↔ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) )
26	25	pm5.32da	⊢ ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) → ( ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∧ 𝐵 ∈ ( 𝐴 × V ) ) ↔ ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∧ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) ) )
27		simpr	⊢ ( ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∧ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) → 𝐵 ∈ ( 𝐴 × 𝐴 ) )
28		1st2nd2	⊢ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) → 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ )
29	28	ancri	⊢ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) → ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∧ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) )
30	27 29	impbii	⊢ ( ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∧ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) ↔ 𝐵 ∈ ( 𝐴 × 𝐴 ) )
31	26 30	bitrdi	⊢ ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) → ( ( 𝐵 = ⟨ ( 1^st ‘ 𝐵 ) , ( 2^nd ‘ 𝐵 ) ⟩ ∧ 𝐵 ∈ ( 𝐴 × V ) ) ↔ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) )
32	7 31	syl5bb	⊢ ( ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) → ( 𝐵 ∈ ( 𝐴 × V ) ↔ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) )
33	32	pm5.32ri	⊢ ( ( 𝐵 ∈ ( 𝐴 × V ) ∧ ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ) ↔ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ∧ ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ) )
34	3 5 33	3bitri	⊢ ( 𝐵 ∈ ( I ↾ 𝐴 ) ↔ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ∧ ( 1^st ‘ 𝐵 ) = ( 2^nd ‘ 𝐵 ) ) )

Metamath Proof Explorer

Theorem bj-elid6

Proof