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Theorem idref 5757
Description: TODO: This is the same as issref 5011 (which has a much longer proof). Should we replace issref 5011 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 2177 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
21fmpt 5666 . . 3 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅)
3 vex 2740 . . . . . 6 𝑥 ∈ V
43, 3opex 4229 . . . . 5 𝑥, 𝑥⟩ ∈ V
54, 1fnmpti 5344 . . . 4 (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
6 df-f 5220 . . . 4 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
75, 6mpbiran 940 . . 3 ((𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
82, 7bitri 184 . 2 (∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
9 df-br 4004 . . 3 (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
109ralbii 2483 . 2 (∀𝑥𝐴 𝑥𝑅𝑥 ↔ ∀𝑥𝐴𝑥, 𝑥⟩ ∈ 𝑅)
11 mptresid 4961 . . . 4 (𝑥𝐴𝑥) = ( I ↾ 𝐴)
123fnasrn 5694 . . . 4 (𝑥𝐴𝑥) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1311, 12eqtr3i 2200 . . 3 ( I ↾ 𝐴) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩)
1413sseq1i 3181 . 2 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
158, 10, 143bitr4ri 213 1 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2148  wral 2455  wss 3129  cop 3595   class class class wbr 4003  cmpt 4064   I cid 4288  ran crn 4627  cres 4628   Fn wfn 5211  wf 5212
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224
This theorem is referenced by: (None)
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