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Theorem resiexg 7326
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 6698). (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 5623 . . 3 Rel ( I ↾ 𝐴)
2 simpr 473 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
3 eleq1w 2864 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
43biimpa 464 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑦𝐴)
52, 4jca 503 . . . 4 ((𝑥 = 𝑦𝑥𝐴) → (𝑥𝐴𝑦𝐴))
6 vex 3390 . . . . . 6 𝑦 ∈ V
76opelres 5599 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴))
8 df-br 4838 . . . . . . 7 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
96ideq 5470 . . . . . . 7 (𝑥 I 𝑦𝑥 = 𝑦)
108, 9bitr3i 268 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1110anbi1i 612 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
127, 11bitri 266 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
13 opelxp 5340 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
145, 12, 133imtr4i 283 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
151, 14relssi 5407 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
16 sqxpexg 7187 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
17 ssexg 4993 . 2 ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
1815, 16, 17sylancr 577 1 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 2155  Vcvv 3387  wss 3763  cop 4370   class class class wbr 4837   I cid 5212   × cxp 5303  cres 5307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-br 4838  df-opab 4900  df-id 5213  df-xp 5311  df-rel 5312  df-res 5317
This theorem is referenced by:  ordiso  8654  wdomref  8710  dfac9  9237  relexp0g  13979  relexpsucnnr  13982  ndxarg  16087  idfu2nd  16735  idfu1st  16737  idfucl  16739  setcid  16934  equivestrcsetc  16991  pf1ind  19921  islinds2  20356  ausgrusgrb  26269  upgrres1lem1  26411  cusgrexilem1  26557  sizusglecusg  26581  pliguhgr  27663  bj-evalid  33333  poimirlem15  33731  xrnidresex  34472  dib0  36939  dicn0  36967  cdlemn11a  36982  dihord6apre  37031  dihatlat  37109  dihpN  37111  eldioph2lem1  37819  eldioph2lem2  37820  dfrtrcl5  38430  dfrcl2  38460  relexpiidm  38490  uspgrsprfo  42318  rngcidALTV  42553  ringcidALTV  42616
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