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Theorem idref 7137
Description: Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.)
Assertion
Ref Expression
idref (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem idref
StepHypRef Expression
1 eqid 2724 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
21fmpt 7102 . . 3 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅)
3 opex 5455 . . . . 5 𝑥, 𝑥⟩ ∈ V
43, 1fnmpti 6684 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
5 df-f 6538 . . . 4 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
64, 5mpbiran 706 . . 3 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
72, 6bitri 275 . 2 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
8 df-br 5140 . . 3 (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
98ralbii 3085 . 2 (∀𝑥𝐴 𝑥𝑅𝑥 ↔ ∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅)
10 mptresid 6041 . . . 4 ( I ↾ 𝐴) = (𝑥𝐴𝑥)
11 vex 3470 . . . . 5 𝑥 ∈ V
1211fnasrn 7136 . . . 4 (𝑥𝐴𝑥) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1310, 12eqtri 2752 . . 3 ( I ↾ 𝐴) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1413sseq1i 4003 . 2 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
157, 9, 143bitr4ri 304 1 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  wral 3053  wss 3941  cop 4627   class class class wbr 5139  cmpt 5222   I cid 5564  ran crn 5668  cres 5669   Fn wfn 6529  wf 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541
This theorem is referenced by:  retos  21500  filnetlem2  35765
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