Theorem cnvref5 37220

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 5787	. 2 ⊢ (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 I 𝑦)))
2		ideqg 5852	. . . . 5 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
3	2	elv 3481	. . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4	3	imbi2i 336	. . 3 ⊢ ((𝑥𝑅𝑦 → 𝑥 I 𝑦) ↔ (𝑥𝑅𝑦 → 𝑥 = 𝑦))
5	4	2albii 1823	. 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 205  ∀wal 1540   = wceq 1542  Vcvv 3475   ⊆ wss 3949   class class class wbr 5149   I cid 5574  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684

This theorem is referenced by:  dfcnvrefrel5  37403