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Theorem resiexg 7428
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 6798). (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 5754 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
2 sqxpexg 7288 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
3 ssexg 5077 . 2 ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
41, 2, 3sylancr 578 1 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2048  Vcvv 3409  wss 3825   I cid 5304   × cxp 5398  cres 5402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-id 5305  df-xp 5406  df-rel 5407  df-res 5412
This theorem is referenced by:  ordiso  8767  wdomref  8823  dfac9  9348  relexp0g  14232  relexpsucnnr  14235  ndxarg  16354  idfu2nd  16995  idfu1st  16997  idfucl  16999  funcestrcsetclem4  17241  equivestrcsetc  17250  funcsetcestrclem4  17256  pf1ind  20210  islinds2  20649  ausgrusgrb  26643  upgrres1lem1  26784  cusgrexilem1  26914  sizusglecusg  26938  pliguhgr  28030  bj-evalid  33811  poimirlem15  34296  xrnidresex  35048  dib0  37693  dicn0  37721  cdlemn11a  37736  dihord6apre  37785  dihatlat  37863  dihpN  37865  eldioph2lem1  38697  eldioph2lem2  38698  dfrtrcl5  39297  dfrcl2  39327  relexpiidm  39357  uspgrsprfo  43331  rngcidALTV  43566  ringcidALTV  43629
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