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Theorem inxpss3 38272
Description: Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss 38269). (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 5777 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦))
2 brinxp2 5777 . . . . 5 (𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑆𝑦))
31, 2imbi12i 350 . . . 4 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑆𝑦)))
4 imdistan 567 . . . 4 (((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) → ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑆𝑦)))
53, 4bitr4i 278 . . 3 ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
652albii 1818 . 2 (∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
7 r2al 3201 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → (𝑥𝑅𝑦𝑥𝑆𝑦)))
86, 7bitr4i 278 1 (∀𝑥𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥𝑆𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wcel 2108  wral 3067  cin 3975   class class class wbr 5166   × cxp 5698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706
This theorem is referenced by:  inxpss2  38273
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