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Theorem resiexg 7934
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7235). (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 6067 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2 sqxpexg 7775 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
3 ssexg 5323 . 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 2108  Vcvv 3480  wss 3951   I cid 5577   × cxp 5683  cres 5687
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-res 5697
This theorem is referenced by:  ordiso  9556  wdomref  9612  dfac9  10177  relexp0g  15061  relexpsucnnr  15064  ndxarg  17233  idfu2nd  17922  idfu1st  17924  idfucl  17926  funcestrcsetclem4  18188  equivestrcsetc  18197  funcsetcestrclem4  18203  sursubmefmnd  18909  injsubmefmnd  18910  smndex1n0mnd  18925  islinds2  21833  pf1ind  22359  ausgrusgrb  29182  upgrres1lem1  29326  cusgrexilem1  29456  sizusglecusg  29481  pliguhgr  30505  bj-evalid  37077  bj-diagval  37175  poimirlem15  37642  xrnidresex  38408  dib0  41166  dicn0  41194  cdlemn11a  41209  dihord6apre  41258  dihatlat  41336  dihpN  41338  eldioph2lem1  42771  eldioph2lem2  42772  dfrtrcl5  43642  dfrcl2  43687  relexpiidm  43717  ushggricedg  47896  uspgrsprfo  48064  rngcidALTV  48190  ringcidALTV  48224
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