Theorem dirref 17235

Description: A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Hypothesis

Ref	Expression
dirref.1	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
dirref	⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

Proof of Theorem dirref

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 ⊢ 𝐴 = 𝐴
2		resieq 5407	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴))
3	2	anidms 677	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴))
4	1, 3	mpbiri 248	. . 3 ⊢ (𝐴 ∈ 𝑋 → 𝐴( I ↾ 𝑋)𝐴)
5		dirref.1	. . . . . . 7 ⊢ 𝑋 = dom 𝑅
6		dirdm 17234	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)
7	5, 6	syl5eq 2668	. . . . . 6 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅)
8	7	reseq2d 5396	. . . . 5 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ ∪ ∪ 𝑅))
9		eqid 2622	. . . . . . . . 9 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
10	9	isdir 17232	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
11	10	ibi 256	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
12	11	simpld 475	. . . . . 6 ⊢ (𝑅 ∈ DirRel → (Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅))
13	12	simprd 479	. . . . 5 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅)
14	8, 13	eqsstrd 3639	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
15	14	ssbrd 4696	. . 3 ⊢ (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴 → 𝐴𝑅𝐴))
16	4, 15	syl5 34	. 2 ⊢ (𝑅 ∈ DirRel → (𝐴 ∈ 𝑋 → 𝐴𝑅𝐴))
17	16	imp 445	1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∪ cuni 4436   class class class wbr 4653   I cid 5023   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116   ∘ ccom 5118  Rel wrel 5119  DirRelcdir 17228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-dir 17230

This theorem is referenced by:  tailini  32371