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Theorem inxpss3 37818
Description: Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss 37815). (Contributed by Peter Mazsa, 8-Mar-2019.)
Assertion
Ref Expression
inxpss3 (∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)

Proof of Theorem inxpss3
StepHypRef Expression
1 brinxp2 5759 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦))
2 brinxp2 5759 . . . . 5 (𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑆𝑦))
31, 2imbi12i 349 . . . 4 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑆𝑦)))
4 imdistan 566 . . . 4 (((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑆𝑦)))
53, 4bitr4i 277 . . 3 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
652albii 1814 . 2 (∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
7 r2al 3192 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
86, 7bitr4i 277 1 (∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wcel 2098  wral 3058  cin 3948   class class class wbr 5152   × cxp 5680
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688
This theorem is referenced by:  inxpss2  37819
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