Theorem dfsymrel2 38592

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 38587	. 2 ⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		dfrel6 38381	. . . . . 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 5815	. . . 4 ⊢ (Rel 𝑅 → ◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ◡𝑅)
5	4, 3	sseq12d 3968	. . 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 3901   ⊆ wss 3902   × cxp 5614  ^◡ccnv 5615  dom cdm 5616  ran crn 5617  Rel wrel 5621   SymRel wsymrel 38233

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-symrel 38587

This theorem is referenced by:  dfsymrel3  38593  dfsymrel4  38594  dfsymrel5  38595  elsymrelsrel  38600  symreleq  38601  symrelcoss  38603  refsymrel2  38610  eqvrelsym  38648  refrelredund4  38678