Theorem refsymrel2 39162

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 39106, cf. the comment of dfrefrels2 39104. (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 39106	. . . 4 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2		dfsymrel2 39144	. . . 4 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
3	1, 2	anbi12i 639	. . 3 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)))
4		anandi3r 1118	. . 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 1111	. . 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 302	. 2 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
7		symrefref2 39158	. . . 4 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
8	7	pm5.32ri 585	. . 3 ⊢ ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅))
9	8	anbi1i 635	. 2 ⊢ (((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
10	6, 9	bitri 278	1 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   ∩ cin 3906   ⊆ wss 3907   I cid 5546   × cxp 5650  ^◡ccnv 5651  dom cdm 5652  ran crn 5653   ↾ cres 5654  Rel wrel 5657   RefRel wrefrel 38700   SymRel wsymrel 38706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-refrel 39103  df-symrel 39135

This theorem is referenced by:  dfeqvrel2  39185  refrelredund4  39230