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Theorem inxpssidinxp 36013
 Description: Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 36012. (Contributed by Peter Mazsa, 4-Jul-2019.)
Assertion
Ref Expression
inxpssidinxp ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem inxpssidinxp
StepHypRef Expression
1 inxpss2 36012 . 2 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 I 𝑦))
2 ideqg 5691 . . . . 5 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3415 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
43imbi2i 339 . . 3 ((𝑥𝑅𝑦𝑥 I 𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦))
542ralbii 3098 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 I 𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
61, 5bitri 278 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wral 3070  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858   class class class wbr 5032   I cid 5429   × cxp 5522 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-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 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 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531 This theorem is referenced by:  dfcnvrefrels3  36207  dfcnvrefrel3  36209
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