Theorem refsymrel2 38820

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 38764, cf. the comment of dfrefrels2 38762. (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 38764	. . . 4 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2		dfsymrel2 38802	. . . 4 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
3	1, 2	anbi12i 628	. . 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 38816	. . . 4 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
8	7	pm5.32ri 575	. . 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 275	1 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∩ cin 3900   ⊆ wss 3901   I cid 5518   × cxp 5622  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ↾ cres 5626  Rel wrel 5629   RefRel wrefrel 38385   SymRel wsymrel 38391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-refrel 38761  df-symrel 38793

This theorem is referenced by:  dfeqvrel2  38843  refrelredund4  38888