Theorem idref 5956

Description: TODO: This is the same as issref 5168 (which has a much longer proof). Should we replace issref 5168 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 2238	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
2	1	fmpt 5852	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅)
3		vex 2824	. . . . . 6 ⊢ 𝑥 ∈ V
4	3, 3	opex 4367	. . . . 5 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
5	4, 1	fnmpti 5510	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
6		df-f 5379	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
7	5, 6	mpbiran 953	. . 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 4129	. . 3 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
10	9	ralbii 2556	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅)
11		mptresid 5115	. . . . 5 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)
12	11	eqcomi 2242	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ( I ↾ 𝐴)
13	3	fnasrn 5881	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
14	12, 13	eqtr3i 2261	. . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
15	14	sseq1i 3274	. 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 2209  ∀wral 2528   ⊆ wss 3220  ⟨cop 3711   class class class wbr 4128   ↦ cmpt 4190   I cid 4431  ran crn 4773   ↾ cres 4774   Fn wfn 5370  ⟶wf 5371

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383

This theorem is referenced by: (None)