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Theorem idref 6890
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 2822 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
21fmpt 6856 . . 3 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅)
3 opex 5333 . . . . 5 𝑥, 𝑥⟩ ∈ V
43, 1fnmpti 6471 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
5 df-f 6338 . . . 4 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
64, 5mpbiran 708 . . 3 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
72, 6bitri 278 . 2 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
8 df-br 5043 . . 3 (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
98ralbii 3157 . 2 (∀𝑥𝐴 𝑥𝑅𝑥 ↔ ∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅)
10 mptresid 5896 . . . 4 ( I ↾ 𝐴) = (𝑥𝐴𝑥)
11 vex 3472 . . . . 5 𝑥 ∈ V
1211fnasrn 6889 . . . 4 (𝑥𝐴𝑥) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1310, 12eqtri 2845 . . 3 ( I ↾ 𝐴) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1413sseq1i 3970 . 2 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
157, 9, 143bitr4ri 307 1 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2114  wral 3130  wss 3908  cop 4545   class class class wbr 5042  cmpt 5122   I cid 5436  ran crn 5533  cres 5534   Fn wfn 6329  wf 6330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342
This theorem is referenced by:  retos  20305  filnetlem2  33801
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