Theorem bj-elid3 33087

Description: Characterization of the elements of I. (Contributed by BJ, 29-Mar-2020.)

Assertion

Ref	Expression
bj-elid3	⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-elid3

Step	Hyp	Ref	Expression
1		bj-elid 33085	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
2		opelxp 5146	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	anbi1i 731	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
4		op1stg 7180	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
5		op2ndg 7181	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
6	4, 5	eqeq12d 2637	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩) ↔ 𝐴 = 𝐵))
7	6	pm5.32i 669	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
8		simpl 473	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
9	8	anim1i 592	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐴 = 𝐵))
10		simpl 473	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
11		eleq1 2689	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
12	11	biimpac 503	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
13		simpr 477	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
14	10, 12, 13	jca31 557	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
15	9, 14	impbii 199	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
16	7, 15	bitri 264	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
17	3, 16	bitri 264	. 2 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
18	1, 17	bitri 264	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   I cid 5023   × cxp 5112  ‘cfv 5888  1^st c1st 7166  2^nd c2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  bj-eldiag2  33092