Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inxpssidinxp Structured version   Visualization version   GIF version

Theorem inxpssidinxp 35014
 Description: Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 35013. (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 35013 . 2 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 I 𝑦))
2 ideqg 5572 . . . . 5 (𝑦 ∈ V → (𝑥 I 𝑦𝑥 = 𝑦))
32elv 3421 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
43imbi2i 328 . . 3 ((𝑥𝑅𝑦𝑥 I 𝑦) ↔ (𝑥𝑅𝑦𝑥 = 𝑦))
542ralbii 3117 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 I 𝑦) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
61, 5bitri 267 1 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wral 3089  Vcvv 3416   ∩ cin 3829   ⊆ wss 3830   class class class wbr 4929   I cid 5311   × cxp 5405 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414 This theorem is referenced by:  dfcnvrefrels3  35209  dfcnvrefrel3  35211
 Copyright terms: Public domain W3C validator