Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfsymrel2 Structured version   Visualization version   GIF version

Theorem dfsymrel2 37057
Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
Assertion
Ref Expression
dfsymrel2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))

Proof of Theorem dfsymrel2
StepHypRef Expression
1 df-symrel 37052 . 2 ( SymRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2 dfrel6 36854 . . . . . 6 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32biimpi 215 . . . . 5 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
43cnveqd 5832 . . . 4 (Rel 𝑅(𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
54, 3sseq12d 3978 . . 3 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅𝑅))
65pm5.32ri 577 . 2 (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (𝑅𝑅 ∧ Rel 𝑅))
71, 6bitri 275 1 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  cin 3910  wss 3911   × cxp 5632  ccnv 5633  dom cdm 5634  ran crn 5635  Rel wrel 5639   SymRel wsymrel 36692
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-symrel 37052
This theorem is referenced by:  dfsymrel3  37058  dfsymrel4  37059  dfsymrel5  37060  elsymrelsrel  37065  symreleq  37066  symrelcoss  37068  refsymrel2  37075  eqvrelsym  37113  refrelredund4  37143
  Copyright terms: Public domain W3C validator