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Theorem idinxpssinxp 36991
Description: Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 36989. (Contributed by Peter Mazsa, 6-Mar-2019.)
Assertion
Ref Expression
idinxpssinxp (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem idinxpssinxp
StepHypRef Expression
1 inxpss2 36989 . 2 (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 I 𝑦𝑥𝑅𝑦))
2 ideqg 5843 . . . . 5 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3479 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
43imbi1i 349 . . 3 ((𝑥 I 𝑦𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑥𝑅𝑦))
542ralbii 3127 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥 I 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
61, 5bitri 274 1 (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wral 3060  Vcvv 3473  cin 3943  wss 3944   class class class wbr 5141   I cid 5566   × cxp 5667
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676
This theorem is referenced by:  idinxpssinxp4  36994
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