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Theorem inxpss 35701
Description: Two ways to say that an intersection with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.)
Assertion
Ref Expression
inxpss ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ 𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Proof of Theorem inxpss
StepHypRef Expression
1 brinxp2 5617 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦))
21imbi1i 353 . . . 4 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → 𝑥𝑆𝑦))
3 impexp 454 . . . 4 ((((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → 𝑥𝑆𝑦) ↔ ((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
42, 3bitri 278 . . 3 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦) ↔ ((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
542albii 1822 . 2 (∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
6 relinxp 5675 . . 3 Rel (𝑅 ∩ (𝐴 × 𝐵))
7 ssrel3 35688 . . 3 (Rel (𝑅 ∩ (𝐴 × 𝐵)) → ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ 𝑆 ↔ ∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦)))
86, 7ax-mp 5 . 2 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ 𝑆 ↔ ∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦))
9 r2al 3196 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
105, 8, 93bitr4i 306 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ 𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wcel 2115  wral 3133  cin 3918  wss 3919   class class class wbr 5053   × cxp 5541  Rel wrel 5548
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-xp 5549  df-rel 5550
This theorem is referenced by:  idinxpss  35702
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