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Theorem inxpssidinxp 35454
Description: Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 35453. (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 35453 . 2 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 I 𝑦))
2 ideqg 5715 . . . . 5 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3497 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
43imbi2i 337 . . 3 ((𝑥𝑅𝑦𝑥 I 𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦))
542ralbii 3163 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 I 𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
61, 5bitri 276 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wral 3135  Vcvv 3492  cin 3932  wss 3933   class class class wbr 5057   I cid 5452   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555
This theorem is referenced by:  dfcnvrefrels3  35647  dfcnvrefrel3  35649
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