Theorem refrelredund4 38633

Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 38513) if the relation is symmetric as well. (Contributed by Peter Mazsa, 26-Oct-2022.)

Assertion

Ref	Expression
refrelredund4	⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))

Proof of Theorem refrelredund4

Step	Hyp	Ref	Expression
1		inxpssres 5658	. . . . 5 ⊢ ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ↾ dom 𝑅)
2		sstr2 3956	. . . . 5 ⊢ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ↾ dom 𝑅) → (( I ↾ dom 𝑅) ⊆ 𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅))
3	1, 2	ax-mp 5	. . . 4 ⊢ (( I ↾ dom 𝑅) ⊆ 𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅)
4	3	anim1i 615	. . 3 ⊢ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
5		dfrefrel2 38513	. . 3 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
6	4, 5	sylibr 234	. 2 ⊢ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → RefRel 𝑅)
7		an12 645	. . 3 ⊢ (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅)))
8		anandir 677	. . . . 5 ⊢ (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)))
9		refsymrel2 38565	. . . . 5 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
10		dfsymrel2 38547	. . . . . 6 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
11	10	anbi2i 623	. . . . 5 ⊢ (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)))
12	8, 9, 11	3bitr4i 303	. . . 4 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅))
13	12	anbi2i 623	. . 3 ⊢ (( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ SymRel 𝑅)))
14	7, 13	bitr4i 278	. 2 ⊢ (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)))
15		df-redundp 38623	. 2 ⊢ ( redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) → RefRel 𝑅) ∧ (((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅) ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)) ↔ ( RefRel 𝑅 ∧ ( RefRel 𝑅 ∧ SymRel 𝑅)))))
16	6, 14, 15	mpbir2an 711	1 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∩ cin 3916   ⊆ wss 3917   I cid 5535   × cxp 5639  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   ↾ cres 5643  Rel wrel 5646   RefRel wrefrel 38182   SymRel wsymrel 38188   redund wredundp 38198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-refrel 38510  df-symrel 38542  df-redundp 38623

This theorem is referenced by:  refrelredund2  38634