Theorem bj-elid 32262

Description: Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-elid	⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))

Proof of Theorem bj-elid

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reli 5171	. . . . 5 ⊢ Rel I
2		df-rel 5045	. . . . 5 ⊢ (Rel I ↔ I ⊆ (V × V))
3	1, 2	mpbi 219	. . . 4 ⊢ I ⊆ (V × V)
4	3	sseli 3564	. . 3 ⊢ (𝐴 ∈ I → 𝐴 ∈ (V × V))
5		1st2nd2 7096	. . . . . . 7 ⊢ (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝐴 ∈ I → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
7	6	eleq1d 2672	. . . . 5 ⊢ (𝐴 ∈ I → (𝐴 ∈ I ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ))
8	7	ibi 255	. . . 4 ⊢ (𝐴 ∈ I → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I )
9		df-id 4953	. . . . . 6 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
10	9	eleq2i 2680	. . . . 5 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
11		fvex 6113	. . . . . 6 ⊢ (1st ‘𝐴) ∈ V
12		fvex 6113	. . . . . 6 ⊢ (2nd ‘𝐴) ∈ V
13		eqeq12 2623	. . . . . 6 ⊢ ((𝑥 = (1st ‘𝐴) ∧ 𝑦 = (2nd ‘𝐴)) → (𝑥 = 𝑦 ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
14	11, 12, 13	opelopaba 4916	. . . . 5 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ (1st ‘𝐴) = (2nd ‘𝐴))
15	10, 14	bitri 263	. . . 4 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))
16	8, 15	sylib 207	. . 3 ⊢ (𝐴 ∈ I → (1st ‘𝐴) = (2nd ‘𝐴))
17	4, 16	jca 553	. 2 ⊢ (𝐴 ∈ I → (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))
18	5	eleq1d 2672	. . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ))
19	18	biimprd 237	. . . 4 ⊢ (𝐴 ∈ (V × V) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I → 𝐴 ∈ I ))
20	15, 19	syl5bir 232	. . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = (2nd ‘𝐴) → 𝐴 ∈ I ))
21	20	imp 444	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)) → 𝐴 ∈ I )
22	17, 21	impbii 198	1 ⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131  {copab 4642   I cid 4948   × cxp 5036  Rel wrel 5043  ‘cfv 5804  1^st c1st 7057  2^nd c2nd 7058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060

This theorem is referenced by:  bj-elid2  32263  bj-elid3  32264