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Theorem inxpss2 37674
Description: Two ways to say that intersections with Cartesian products are in a subclass relation. (Contributed by Peter Mazsa, 8-Mar-2019.)
Assertion
Ref Expression
inxpss2 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑆 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Proof of Theorem inxpss2
StepHypRef Expression
1 relinxp 5804 . . 3 Rel (𝑅 ∩ (𝐴 × 𝐵))
2 ssrel3 5776 . . 3 (Rel (𝑅 ∩ (𝐴 × 𝐵)) → ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑆 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦)))
31, 2ax-mp 5 . 2 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑆 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦))
4 inxpss3 37673 . 2 (∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
53, 4bitri 275 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑆 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wral 3053  cin 3939  wss 3940   class class class wbr 5138   × cxp 5664  Rel wrel 5671
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 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
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 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673
This theorem is referenced by:  inxpssidinxp  37675  idinxpssinxp  37676
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