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Theorem resiexg 7905
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7217). (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 6049 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2 sqxpexg 7742 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
3 ssexg 5324 . 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 3475  wss 3949   I cid 5574   × cxp 5675  cres 5679
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-res 5689
This theorem is referenced by:  ordiso  9511  wdomref  9567  dfac9  10131  relexp0g  14969  relexpsucnnr  14972  ndxarg  17129  idfu2nd  17827  idfu1st  17829  idfucl  17831  funcestrcsetclem4  18095  equivestrcsetc  18104  funcsetcestrclem4  18110  sursubmefmnd  18777  injsubmefmnd  18778  smndex1n0mnd  18793  islinds2  21368  pf1ind  21874  ausgrusgrb  28425  upgrres1lem1  28566  cusgrexilem1  28696  sizusglecusg  28720  pliguhgr  29739  bj-evalid  35957  bj-diagval  36055  poimirlem15  36503  xrnidresex  37277  dib0  40035  dicn0  40063  cdlemn11a  40078  dihord6apre  40127  dihatlat  40205  dihpN  40207  eldioph2lem1  41498  eldioph2lem2  41499  dfrtrcl5  42380  dfrcl2  42425  relexpiidm  42455  uspgrsprfo  46526  rngcidALTV  46889  ringcidALTV  46952
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