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Theorem inxpss 34406
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 brinxp2ALTV 34358 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦))
21imbi1i 338 . . . 4 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → 𝑥𝑆𝑦))
3 impexp 461 . . . 4 ((((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → 𝑥𝑆𝑦) ↔ ((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
42, 3bitri 264 . . 3 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦) ↔ ((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
542albii 1897 . 2 (∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
6 relinxp 34393 . . 3 Rel (𝑅 ∩ (𝐴 × 𝐵))
7 ssrel3 34391 . . 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 3077 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
105, 8, 93bitr4i 292 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ 𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wcel 2139  wral 3050  cin 3714  wss 3715   class class class wbr 4804   × cxp 5264  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by:  idinxpss  34407
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