Theorem dfsymrels2 36282

Description: Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 36254. (Contributed by Peter Mazsa, 20-Jul-2019.)

Assertion

Ref	Expression
dfsymrels2	⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}

Proof of Theorem dfsymrels2

Step	Hyp	Ref	Expression
1		df-symrels 36280	. 2 ⊢ SymRels = ( Syms ∩ Rels )
2		df-syms 36279	. 2 ⊢ Syms = {𝑟 ∣ ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		inex1g 5187	. . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
4	3	elv 3404	. . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5		brssr 36242	. . . 4 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟))))
6	4, 5	ax-mp 5	. . 3 ⊢ (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)))
7		elrels6 36231	. . . . . . 7 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
8	7	elv 3404	. . . . . 6 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
9	8	biimpi 219	. . . . 5 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
10	9	cnveqd 5718	. . . 4 ⊢ (𝑟 ∈ Rels → ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) = ◡𝑟)
11	10, 9	sseq12d 3910	. . 3 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟))
12	6, 11	syl5bb 286	. 2 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟))
13	1, 2, 12	abeqinbi 36016	1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}

Syntax hints:   ↔ wb 209   = wceq 1542   ∈ wcel 2114  {crab 3057  Vcvv 3398   ∩ cin 3842   ⊆ wss 3843   class class class wbr 5030   × cxp 5523  ^◡ccnv 5524  dom cdm 5525  ran crn 5526   Rels crels 35958   S cssr 35959   Syms csyms 35966   SymRels csymrels 35967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-rels 36226  df-ssr 36239  df-syms 36279  df-symrels 36280

This theorem is referenced by:  dfsymrels3  36283  dfsymrels4  36284  elsymrels2  36290  refsymrels2  36302  refrelsredund4  36368