Theorem bj-elid 33663

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 5495	. . . . 5 ⊢ Rel I
2		df-rel 5362	. . . . 5 ⊢ (Rel I ↔ I ⊆ (V × V))
3	1, 2	mpbi 222	. . . 4 ⊢ I ⊆ (V × V)
4	3	sseli 3817	. . 3 ⊢ (𝐴 ∈ I → 𝐴 ∈ (V × V))
5		1st2nd2 7484	. . . . . . 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 2844	. . . . 5 ⊢ (𝐴 ∈ I → (𝐴 ∈ I ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ))
8	7	ibi 259	. . . 4 ⊢ (𝐴 ∈ I → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I )
9		df-id 5261	. . . . . 6 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
10	9	eleq2i 2851	. . . . 5 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
11		fvex 6459	. . . . . 6 ⊢ (1st ‘𝐴) ∈ V
12		fvex 6459	. . . . . 6 ⊢ (2nd ‘𝐴) ∈ V
13		eqeq12 2791	. . . . . 6 ⊢ ((𝑥 = (1st ‘𝐴) ∧ 𝑦 = (2nd ‘𝐴)) → (𝑥 = 𝑦 ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
14	11, 12, 13	opelopaba 5228	. . . . 5 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ (1st ‘𝐴) = (2nd ‘𝐴))
15	10, 14	bitri 267	. . . 4 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))
16	8, 15	sylib 210	. . 3 ⊢ (𝐴 ∈ I → (1st ‘𝐴) = (2nd ‘𝐴))
17	4, 16	jca 507	. 2 ⊢ (𝐴 ∈ I → (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))
18	5	eleq1d 2844	. . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I ))
19	18	biimprd 240	. . . 4 ⊢ (𝐴 ∈ (V × V) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ I → 𝐴 ∈ I ))
20	15, 19	syl5bir 235	. . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = (2nd ‘𝐴) → 𝐴 ∈ I ))
21	20	imp 397	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)) → 𝐴 ∈ I )
22	17, 21	impbii 201	1 ⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))

Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ⊆ wss 3792  ⟨cop 4404  {copab 4948   I cid 5260   × cxp 5353  Rel wrel 5360  ‘cfv 6135  1^st c1st 7443  2^nd c2nd 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fv 6143  df-1st 7445  df-2nd 7446

This theorem is referenced by:  bj-elid2  33664  bj-elid3  33665