Theorem elrefsymrelsrel 37980

Description: For sets, being an element of the class of reflexive and symmetric relations is equivalent to satisfying the reflexive and symmetric relation predicates. (Contributed by Peter Mazsa, 23-Aug-2021.)

Assertion

Ref	Expression
elrefsymrelsrel	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))

Proof of Theorem elrefsymrelsrel

Step	Hyp	Ref	Expression
1		elin 3960	. 2 ⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ (𝑅 ∈ RefRels ∧ 𝑅 ∈ SymRels ))
2		elrefrelsrel 37929	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
3		elsymrelsrel 37966	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
4	2, 3	anbi12d 630	. 2 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ∈ RefRels ∧ 𝑅 ∈ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))
5	1, 4	bitrid 283	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099   ∩ cin 3943   RefRels crefrels 37588   RefRel wrefrel 37589   SymRels csymrels 37594   SymRel wsymrel 37595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-rels 37894  df-ssr 37907  df-refs 37919  df-refrels 37920  df-refrel 37921  df-syms 37951  df-symrels 37952  df-symrel 37953

This theorem is referenced by: (None)