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Theorem idref 6539
Description: TODO: This is the same as issref 5544 (which has a much longer proof). Should we replace issref 5544 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 2651 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
21fmpt 6421 . . 3 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅)
3 opex 4962 . . . . 5 𝑥, 𝑥⟩ ∈ V
43, 1fnmpti 6060 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
5 df-f 5930 . . . 4 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
64, 5mpbiran 973 . . 3 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
72, 6bitri 264 . 2 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
8 df-br 4686 . . 3 (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
98ralbii 3009 . 2 (∀𝑥𝐴 𝑥𝑅𝑥 ↔ ∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅)
10 mptresid 5491 . . . 4 (𝑥𝐴𝑥) = ( I ↾ 𝐴)
11 vex 3234 . . . . 5 𝑥 ∈ V
1211fnasrn 6451 . . . 4 (𝑥𝐴𝑥) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1310, 12eqtr3i 2675 . . 3 ( I ↾ 𝐴) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1413sseq1i 3662 . 2 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
157, 9, 143bitr4ri 293 1 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2030  wral 2941  wss 3607  cop 4216   class class class wbr 4685  cmpt 4762   I cid 5052  ran crn 5144  cres 5145   Fn wfn 5921  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  retos  20012  filnetlem2  32499
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