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Theorem idref 5321
 Description: TODO: This is the same as issref 4634 (which has a much longer proof). Should we replace issref 4634 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) ⊆ 𝑅x A x𝑅x)
Distinct variable groups:   x,A   x,𝑅

Proof of Theorem idref
StepHypRef Expression
1 eqid 2022 . . . 4 (x A ↦ ⟨x, x⟩) = (x A ↦ ⟨x, x⟩)
21fmpt 5244 . . 3 (x Ax, x 𝑅 ↔ (x A ↦ ⟨x, x⟩):A𝑅)
3 vex 2538 . . . . . 6 x V
43, 3opex 3940 . . . . 5 x, x V
54, 1fnmpti 4953 . . . 4 (x A ↦ ⟨x, x⟩) Fn A
6 df-f 4833 . . . 4 ((x A ↦ ⟨x, x⟩):A𝑅 ↔ ((x A ↦ ⟨x, x⟩) Fn A ran (x A ↦ ⟨x, x⟩) ⊆ 𝑅))
75, 6mpbiran 835 . . 3 ((x A ↦ ⟨x, x⟩):A𝑅 ↔ ran (x A ↦ ⟨x, x⟩) ⊆ 𝑅)
82, 7bitri 173 . 2 (x Ax, x 𝑅 ↔ ran (x A ↦ ⟨x, x⟩) ⊆ 𝑅)
9 df-br 3739 . . 3 (x𝑅x ↔ ⟨x, x 𝑅)
109ralbii 2308 . 2 (x A x𝑅xx Ax, x 𝑅)
11 mptresid 4587 . . . 4 (x Ax) = ( I ↾ A)
123fnasrn 5266 . . . 4 (x Ax) = ran (x A ↦ ⟨x, x⟩)
1311, 12eqtr3i 2044 . . 3 ( I ↾ A) = ran (x A ↦ ⟨x, x⟩)
1413sseq1i 2946 . 2 (( I ↾ A) ⊆ 𝑅 ↔ ran (x A ↦ ⟨x, x⟩) ⊆ 𝑅)
158, 10, 143bitr4ri 202 1 (( I ↾ A) ⊆ 𝑅x A x𝑅x)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1374  ∀wral 2284   ⊆ wss 2894  ⟨cop 3353   class class class wbr 3738   ↦ cmpt 3792   I cid 3999  ran crn 4273   ↾ cres 4274   Fn wfn 4824  ⟶wf 4825 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837 This theorem is referenced by: (None)
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