Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idinxpssinxp4 Structured version   Visualization version   GIF version

Theorem idinxpssinxp4 36010
 Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 36008). (Contributed by Peter Mazsa, 8-Mar-2019.)
Assertion
Ref Expression
idinxpssinxp4 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦

Proof of Theorem idinxpssinxp4
StepHypRef Expression
1 idinxpssinxp 36007 . 2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦))
2 idinxpssinxp2 36008 . 2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
31, 2bitr3i 280 1 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wral 3071   ∩ cin 3858   ⊆ wss 3859   class class class wbr 5033   I cid 5430   × cxp 5523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-res 5537 This theorem is referenced by:  refrelcoss3  36136
 Copyright terms: Public domain W3C validator