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Theorem idref 5626
Description: TODO: This is the same as issref 4891 (which has a much longer proof). Should we replace issref 4891 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 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem idref
StepHypRef Expression
1 eqid 2117 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
21fmpt 5538 . . 3 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅)
3 vex 2663 . . . . . 6 𝑥 ∈ V
43, 3opex 4121 . . . . 5 𝑥, 𝑥⟩ ∈ V
54, 1fnmpti 5221 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
6 df-f 5097 . . . 4 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
75, 6mpbiran 909 . . 3 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
82, 7bitri 183 . 2 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
9 df-br 3900 . . 3 (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
109ralbii 2418 . 2 (∀𝑥𝐴 𝑥𝑅𝑥 ↔ ∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅)
11 mptresid 4843 . . . 4 (𝑥𝐴𝑥) = ( I ↾ 𝐴)
123fnasrn 5566 . . . 4 (𝑥𝐴𝑥) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1311, 12eqtr3i 2140 . . 3 ( I ↾ 𝐴) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1413sseq1i 3093 . 2 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
158, 10, 143bitr4ri 212 1 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1465  wral 2393  wss 3041  cop 3500   class class class wbr 3899  cmpt 3959   I cid 4180  ran crn 4510  cres 4511   Fn wfn 5088  wf 5089
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by: (None)
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