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Theorem inxpssidinxp 34428
Description: Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 34427. (Contributed by Peter Mazsa, 4-Jul-2019.)
Assertion
Ref Expression
inxpssidinxp ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem inxpssidinxp
StepHypRef Expression
1 inxpss2 34427 . 2 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 I 𝑦))
2 ideqg 5429 . . . . 5 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 34327 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
43imbi2i 325 . . 3 ((𝑥𝑅𝑦𝑥 I 𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦))
542ralbii 3119 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 I 𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
61, 5bitri 264 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wral 3050  Vcvv 3340  cin 3714  wss 3715   class class class wbr 4804   I cid 5173   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273
This theorem is referenced by:  dfcnvrefrels3  34618  dfcnvrefrel3  34620
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