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Theorem idinxpssinxp 35885
 Description: Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 35883. (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 35883 . 2 (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 I 𝑦𝑥𝑅𝑦))
2 ideqg 5690 . . . . 5 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3447 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
43imbi1i 353 . . 3 ((𝑥 I 𝑦𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑥𝑅𝑦))
542ralbii 3134 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥 I 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
61, 5bitri 278 1 (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wral 3106  Vcvv 3442   ∩ cin 3882   ⊆ wss 3883   class class class wbr 5034   I cid 5428   × cxp 5521 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530 This theorem is referenced by:  idinxpssinxp4  35888
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