Theorem elid 5834

Description: Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 5833 after elrid 5695. (Contributed by BJ, 28-Aug-2022.)

Assertion

Ref	Expression
elid	⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)

Distinct variable group:   𝑥,𝐴

Proof of Theorem elid

Step	Hyp	Ref	Expression
1		reli 5483	. . . . 5 ⊢ Rel I
2		dfrel3 5833	. . . . 5 ⊢ (Rel I ↔ ( I ↾ V) = I )
3	1, 2	mpbi 222	. . . 4 ⊢ ( I ↾ V) = I
4	3	eqcomi 2835	. . 3 ⊢ I = ( I ↾ V)
5	4	eleq2i 2899	. 2 ⊢ (𝐴 ∈ I ↔ 𝐴 ∈ ( I ↾ V))
6		elrid 5695	. 2 ⊢ (𝐴 ∈ ( I ↾ V) ↔ ∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩)
7		rexv 3438	. 2 ⊢ (∃𝑥 ∈ V 𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
8	5, 6, 7	3bitri 289	1 ⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 198   = wceq 1658  ∃wex 1880   ∈ wcel 2166  ∃wrex 3119  Vcvv 3415  ⟨cop 4404   I cid 5250   ↾ cres 5345  Rel wrel 5348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-res 5355

This theorem is referenced by: (None)