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Theorem cnvref5 37732
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 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem cnvref5
StepHypRef Expression
1 ssrel3 5779 . 2 (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑥 I 𝑦)))
2 ideqg 5844 . . . . 5 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3474 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
43imbi2i 336 . . 3 ((𝑥𝑅𝑦𝑥 I 𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦))
542albii 1814 . 2 (∀𝑥𝑦(𝑥𝑅𝑦𝑥 I 𝑦) ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑥 = 𝑦))
61, 5bitrdi 287 1 (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  Vcvv 3468  wss 3943   class class class wbr 5141   I cid 5566  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676
This theorem is referenced by:  dfcnvrefrel5  37915
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