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Theorem inxpss 34418
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 34370 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦))
21imbi1i 338 . . . 4 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → 𝑥𝑆𝑦))
3 impexp 437 . . . 4 ((((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → 𝑥𝑆𝑦) ↔ ((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
42, 3bitri 264 . . 3 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦) ↔ ((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
542albii 1895 . 2 (∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥𝑆𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
6 relinxp 34405 . . 3 Rel (𝑅 ∩ (𝐴 × 𝐵))
7 ssrel3 34403 . . 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 3087 . 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 382  wal 1628  wcel 2144  wral 3060  cin 3720  wss 3721   class class class wbr 4784   × cxp 5247  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256
This theorem is referenced by:  idinxpss  34419
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