Theorem cnvref5 38353

Description: Two ways to say that a relation is a subclass of the identity relation. (Contributed by Peter Mazsa, 26-Jun-2019.)

Assertion

Ref	Expression
cnvref5	⊢ (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem cnvref5

Step	Hyp	Ref	Expression
1		ssrel3 5795	. 2 ⊢ (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 I 𝑦)))
2		ideqg 5861	. . . . 5 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
3	2	elv 3484	. . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4	3	imbi2i 336	. . 3 ⊢ ((𝑥𝑅𝑦 → 𝑥 I 𝑦) ↔ (𝑥𝑅𝑦 → 𝑥 = 𝑦))
5	4	2albii 1819	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 I 𝑦) ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦))
6	1, 5	bitrdi 287	1 ⊢ (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537   = wceq 1539  Vcvv 3479   ⊆ wss 3950   class class class wbr 5142   I cid 5576  Rel wrel 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691

This theorem is referenced by:  dfcnvrefrel5  38535