Theorem dfsymrel2 35789

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 35784	. 2 ⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		dfrel6 35608	. . . . . 6 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
3	2	biimpi 218	. . . . 5 ⊢ (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
4	3	cnveqd 5749	. . . 4 ⊢ (Rel 𝑅 → ◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ◡𝑅)
5	4, 3	sseq12d 4003	. . 3 ⊢ (Rel 𝑅 → (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ ◡𝑅 ⊆ 𝑅))
6	5	pm5.32ri 578	. 2 ⊢ ((◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
7	1, 6	bitri 277	1 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1536   ∩ cin 3938   ⊆ wss 3939   × cxp 5556  ^◡ccnv 5557  dom cdm 5558  ran crn 5559  Rel wrel 5563   SymRel wsymrel 35469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-symrel 35784

This theorem is referenced by:  dfsymrel3  35790  dfsymrel4  35791  dfsymrel5  35792  elsymrelsrel  35797  symreleq  35798  symrelcoss  35800  refsymrel2  35807  eqvrelsym  35844  refrelredund4  35874