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Theorem idref 5337
Description: TODO: This is the same as issref 4649 (which has a much longer proof). Should we replace issref 4649 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 2037 . . . 4 (x A ↦ ⟨x, x⟩) = (x A ↦ ⟨x, x⟩)
21fmpt 5260 . . 3 (x Ax, x 𝑅 ↔ (x A ↦ ⟨x, x⟩):A𝑅)
3 vex 2554 . . . . . 6 x V
43, 3opex 3956 . . . . 5 x, x V
54, 1fnmpti 4968 . . . 4 (x A ↦ ⟨x, x⟩) Fn A
6 df-f 4848 . . . 4 ((x A ↦ ⟨x, x⟩):A𝑅 ↔ ((x A ↦ ⟨x, x⟩) Fn A ran (x A ↦ ⟨x, x⟩) ⊆ 𝑅))
75, 6mpbiran 846 . . 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 3755 . . 3 (x𝑅x ↔ ⟨x, x 𝑅)
109ralbii 2324 . 2 (x A x𝑅xx Ax, x 𝑅)
11 mptresid 4602 . . . 4 (x Ax) = ( I ↾ A)
123fnasrn 5282 . . . 4 (x Ax) = ran (x A ↦ ⟨x, x⟩)
1311, 12eqtr3i 2059 . . 3 ( I ↾ A) = ran (x A ↦ ⟨x, x⟩)
1413sseq1i 2963 . 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 1390  wral 2300  wss 2911  cop 3369   class class class wbr 3754  cmpt 3808   I cid 4015  ran crn 4288  cres 4289   Fn wfn 4839  wf 4840
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852
This theorem is referenced by: (None)
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