Theorem refsymrel2 36681

Description: A relation which is reflexive and symmetric (like an equivalence relation) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrel2 36633, cf. the comment of dfrefrels2 36631. (Contributed by Peter Mazsa, 23-Aug-2021.)

Assertion

Ref	Expression
refsymrel2	⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))

Proof of Theorem refsymrel2

Step	Hyp	Ref	Expression
1		dfrefrel2 36633	. . . 4 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2		dfsymrel2 36663	. . . 4 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
3	1, 2	anbi12i 627	. . 3 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)))
4		anandi3r 1102	. . 3 ⊢ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)))
5		3anan32 1096	. . 3 ⊢ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
6	3, 4, 5	3bitr2i 299	. 2 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
7		symrefref2 36677	. . . 4 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
8	7	pm5.32ri 576	. . 3 ⊢ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅))
9	8	anbi1i 624	. 2 ⊢ (((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
10	6, 9	bitri 274	1 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   ∩ cin 3886   ⊆ wss 3887   I cid 5488   × cxp 5587  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591  Rel wrel 5594   RefRel wrefrel 36339   SymRel wsymrel 36345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-refrel 36630  df-symrel 36658

This theorem is referenced by:  dfeqvrel2  36703  refrelredund4  36748