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Theorem resiexg 7843
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7161). (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 6000 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2 sqxpexg 7681 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
3 ssexg 5278 . 2 ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
41, 2, 3sylancr 587 1 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3443  wss 3908   I cid 5528   × cxp 5629  cres 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-res 5643
This theorem is referenced by:  ordiso  9410  wdomref  9466  dfac9  10030  relexp0g  14861  relexpsucnnr  14864  ndxarg  17022  idfu2nd  17717  idfu1st  17719  idfucl  17721  funcestrcsetclem4  17985  equivestrcsetc  17994  funcsetcestrclem4  18000  sursubmefmnd  18660  injsubmefmnd  18661  smndex1n0mnd  18676  islinds2  21166  pf1ind  21667  ausgrusgrb  27961  upgrres1lem1  28102  cusgrexilem1  28232  sizusglecusg  28256  pliguhgr  29273  bj-evalid  35479  bj-diagval  35577  poimirlem15  36025  xrnidresex  36801  dib0  39559  dicn0  39587  cdlemn11a  39602  dihord6apre  39651  dihatlat  39729  dihpN  39731  eldioph2lem1  40986  eldioph2lem2  40987  dfrtrcl5  41806  dfrcl2  41851  relexpiidm  41881  uspgrsprfo  45945  rngcidALTV  46184  ringcidALTV  46247
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