Theorem dfsymrel2 36400

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 36395	. 2 ⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		dfrel6 36219	. . . . . 6 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
3	2	biimpi 219	. . . . 5 ⊢ (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
4	3	cnveqd 5744	. . . 4 ⊢ (Rel 𝑅 → ◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ◡𝑅)
5	4, 3	sseq12d 3934	. . 3 ⊢ (Rel 𝑅 → (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ ◡𝑅 ⊆ 𝑅))
6	5	pm5.32ri 579	. 2 ⊢ ((◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
7	1, 6	bitri 278	1 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543   ∩ cin 3865   ⊆ wss 3866   × cxp 5549  ^◡ccnv 5550  dom cdm 5551  ran crn 5552  Rel wrel 5556   SymRel wsymrel 36082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-symrel 36395

This theorem is referenced by:  dfsymrel3  36401  dfsymrel4  36402  dfsymrel5  36403  elsymrelsrel  36408  symreleq  36409  symrelcoss  36411  refsymrel2  36418  eqvrelsym  36455  refrelredund4  36485