Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idinxpss Structured version   Visualization version   GIF version

Theorem idinxpss 36036
 Description: Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.)
Assertion
Ref Expression
idinxpss (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem idinxpss
StepHypRef Expression
1 inxpss 36035 . 2 (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 I 𝑦𝑥𝑅𝑦))
2 ideqg 5696 . . . . 5 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3415 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
43imbi1i 353 . . 3 ((𝑥 I 𝑦𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑥𝑅𝑦))
542ralbii 3098 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥 I 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
61, 5bitri 278 1 (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wral 3070  Vcvv 3409   ∩ cin 3859   ⊆ wss 3860   class class class wbr 5035   I cid 5432   × cxp 5525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-id 5433  df-xp 5533  df-rel 5534 This theorem is referenced by:  refrelcoss2  36170  dfrefrels3  36220  dfrefrel3  36222  symrefref3  36266
 Copyright terms: Public domain W3C validator