Theorem idref 5725

Description: TODO: This is the same as issref 4986 (which has a much longer proof). Should we replace issref 4986 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

Assertion

Ref	Expression
idref	⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem idref

Step	Hyp	Ref	Expression
1		eqid 2165	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
2	1	fmpt 5635	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅)
3		vex 2729	. . . . . 6 ⊢ 𝑥 ∈ V
4	3, 3	opex 4207	. . . . 5 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
5	4, 1	fnmpti 5316	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
6		df-f 5192	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
7	5, 6	mpbiran 930	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
8	2, 7	bitri 183	. 2 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
9		df-br 3983	. . 3 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
10	9	ralbii 2472	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅)
11		mptresid 4938	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴)
12	3	fnasrn 5663	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
13	11, 12	eqtr3i 2188	. . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
14	13	sseq1i 3168	. 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
15	8, 10, 14	3bitr4ri 212	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   ↔ wb 104   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116  ⟨cop 3579   class class class wbr 3982   ↦ cmpt 4043   I cid 4266  ran crn 4605   ↾ cres 4606   Fn wfn 5183  ⟶wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by: (None)