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Theorem resiexg 7144
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 6520). (Contributed by NM, 13-Jan-2007.)
Assertion
Ref Expression
resiexg (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)

Proof of Theorem resiexg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5461 . . 3 Rel ( I ↾ 𝐴)
2 simpr 476 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
3 eleq1 2718 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
43biimpa 500 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑦𝐴)
52, 4jca 553 . . . 4 ((𝑥 = 𝑦𝑥𝐴) → (𝑥𝐴𝑦𝐴))
6 vex 3234 . . . . . 6 𝑦 ∈ V
76opelres 5436 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴))
8 df-br 4686 . . . . . . 7 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
96ideq 5307 . . . . . . 7 (𝑥 I 𝑦𝑥 = 𝑦)
108, 9bitr3i 266 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1110anbi1i 731 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
127, 11bitri 264 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
13 opelxp 5180 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
145, 12, 133imtr4i 281 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
151, 14relssi 5245 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
16 sqxpexg 7005 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
17 ssexg 4837 . 2 ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
1815, 16, 17sylancr 696 1 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  Vcvv 3231  wss 3607  cop 4216   class class class wbr 4685   I cid 5052   × cxp 5141  cres 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-res 5155
This theorem is referenced by:  ordiso  8462  wdomref  8518  dfac9  8996  relexp0g  13806  relexpsucnnr  13809  ndxarg  15929  idfu2nd  16584  idfu1st  16586  idfucl  16588  setcid  16783  equivestrcsetc  16839  pf1ind  19767  islinds2  20200  ausgrusgrb  26105  upgrres1lem1  26246  cusgrexilem1  26391  sizusglecusg  26415  pliguhgr  27468  bj-evalid  33153  poimirlem15  33554  xrnidresex  34305  dib0  36770  dicn0  36798  cdlemn11a  36813  dihord6apre  36862  dihatlat  36940  dihpN  36942  eldioph2lem1  37640  eldioph2lem2  37641  dfrtrcl5  38253  dfrcl2  38283  relexpiidm  38313  uspgrsprfo  42081  rngcidALTV  42316  ringcidALTV  42379
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