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Theorem resiexg 7907
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7218). (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 6047 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2 sqxpexg 7744 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
3 ssexg 5322 . 2 ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
41, 2, 3sylancr 585 1 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104  Vcvv 3472  wss 3947   I cid 5572   × cxp 5673  cres 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-res 5687
This theorem is referenced by:  ordiso  9513  wdomref  9569  dfac9  10133  relexp0g  14973  relexpsucnnr  14976  ndxarg  17133  idfu2nd  17831  idfu1st  17833  idfucl  17835  funcestrcsetclem4  18099  equivestrcsetc  18108  funcsetcestrclem4  18114  sursubmefmnd  18813  injsubmefmnd  18814  smndex1n0mnd  18829  islinds2  21587  pf1ind  22094  ausgrusgrb  28692  upgrres1lem1  28833  cusgrexilem1  28963  sizusglecusg  28987  pliguhgr  30006  bj-evalid  36260  bj-diagval  36358  poimirlem15  36806  xrnidresex  37580  dib0  40338  dicn0  40366  cdlemn11a  40381  dihord6apre  40430  dihatlat  40508  dihpN  40510  eldioph2lem1  41800  eldioph2lem2  41801  dfrtrcl5  42682  dfrcl2  42727  relexpiidm  42757  uspgrsprfo  46824  rngcidALTV  46977  ringcidALTV  47040
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