Theorem idref 5892

Description: TODO: This is the same as issref 5117 (which has a much longer proof). Should we replace issref 5117 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 2229	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
2	1	fmpt 5793	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅)
3		vex 2803	. . . . . 6 ⊢ 𝑥 ∈ V
4	3, 3	opex 4319	. . . . 5 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
5	4, 1	fnmpti 5458	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
6		df-f 5328	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
7	5, 6	mpbiran 946	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
8	2, 7	bitri 184	. 2 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
9		df-br 4087	. . 3 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
10	9	ralbii 2536	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅)
11		mptresid 5065	. . . . 5 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)
12	11	eqcomi 2233	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴)
13	3	fnasrn 5821	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
14	12, 13	eqtr3i 2252	. . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
15	14	sseq1i 3251	. 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
16	8, 10, 15	3bitr4ri 213	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   ↔ wb 105   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3198  ⟨cop 3670   class class class wbr 4086   ↦ cmpt 4148   I cid 4383  ran crn 4724   ↾ cres 4725   Fn wfn 5319  ⟶wf 5320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332

This theorem is referenced by: (None)