Theorem dfsymrel2 38668

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 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))

Proof of Theorem dfsymrel2

Step	Hyp	Ref	Expression
1		df-symrel 38659	. 2 ⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		dfrel6 38402	. . . . . 6 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
3	2	biimpi 216	. . . . 5 ⊢ (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
4	3	cnveqd 5821	. . . 4 ⊢ (Rel 𝑅 → ◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ◡𝑅)
5	4, 3	sseq12d 3964	. . 3 ⊢ (Rel 𝑅 → (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ ◡𝑅 ⊆ 𝑅))
6	5	pm5.32ri 575	. 2 ⊢ ((◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
7	1, 6	bitri 275	1 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∩ cin 3897   ⊆ wss 3898   × cxp 5619  ^◡ccnv 5620  dom cdm 5621  ran crn 5622  Rel wrel 5626   SymRel wsymrel 38257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-symrel 38659

This theorem is referenced by:  dfsymrel3  38669  dfsymrel4  38670  dfsymrel5  38671  elsymrelsrel  38676  symreleq  38677  symrelcoss  38679  refsymrel2  38686  eqvrelsym  38724  refrelredund4  38754