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Theorem resiexg 7852
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7166). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
resiexg (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)

Proof of Theorem resiexg
StepHypRef Expression
1 idssxp 6003 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2 sqxpexg 7690 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
3 ssexg 5281 . 2 ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
41, 2, 3sylancr 588 1 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3444  wss 3911   I cid 5531   × cxp 5632  cres 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-res 5646
This theorem is referenced by:  ordiso  9457  wdomref  9513  dfac9  10077  relexp0g  14913  relexpsucnnr  14916  ndxarg  17073  idfu2nd  17768  idfu1st  17770  idfucl  17772  funcestrcsetclem4  18036  equivestrcsetc  18045  funcsetcestrclem4  18051  sursubmefmnd  18711  injsubmefmnd  18712  smndex1n0mnd  18727  islinds2  21235  pf1ind  21737  ausgrusgrb  28158  upgrres1lem1  28299  cusgrexilem1  28429  sizusglecusg  28453  pliguhgr  29470  bj-evalid  35593  bj-diagval  35691  poimirlem15  36139  xrnidresex  36915  dib0  39673  dicn0  39701  cdlemn11a  39716  dihord6apre  39765  dihatlat  39843  dihpN  39845  eldioph2lem1  41126  eldioph2lem2  41127  dfrtrcl5  41989  dfrcl2  42034  relexpiidm  42064  uspgrsprfo  46136  rngcidALTV  46375  ringcidALTV  46438
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