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Theorem resiexg 7056
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 6439). (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 5390 . . 3 Rel ( I ↾ 𝐴)
2 simpr 477 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
3 eleq1 2686 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
43biimpa 501 . . . . 5 ((𝑥 = 𝑦𝑥𝐴) → 𝑦𝐴)
52, 4jca 554 . . . 4 ((𝑥 = 𝑦𝑥𝐴) → (𝑥𝐴𝑦𝐴))
6 vex 3192 . . . . . 6 𝑦 ∈ V
76opelres 5366 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴))
8 df-br 4619 . . . . . . 7 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
96ideq 5239 . . . . . . 7 (𝑥 I 𝑦𝑥 = 𝑦)
108, 9bitr3i 266 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1110anbi1i 730 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
127, 11bitri 264 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) ↔ (𝑥 = 𝑦𝑥𝐴))
13 opelxp 5111 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
145, 12, 133imtr4i 281 . . 3 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
151, 14relssi 5177 . 2 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
16 sqxpexg 6923 . 2 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
17 ssexg 4769 . 2 ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V)
1815, 16, 17sylancr 694 1 (𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Vcvv 3189  wss 3559  cop 4159   class class class wbr 4618   I cid 4989   × cxp 5077  cres 5081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-res 5091
This theorem is referenced by:  ordiso  8372  wdomref  8428  dfac9  8909  relexp0g  13703  relexpsucnnr  13706  ndxarg  15811  idfu2nd  16465  idfu1st  16467  idfucl  16469  setcid  16664  equivestrcsetc  16720  pf1ind  19647  islinds2  20080  ausgrusgrb  25966  upgrres1lem1  26102  cusgrexilem1  26235  sizusglecusg  26259  pliguhgr  27205  poimirlem15  33083  dib0  35960  dicn0  35988  cdlemn11a  36003  dihord6apre  36052  dihatlat  36130  dihpN  36132  eldioph2lem1  36830  eldioph2lem2  36831  dfrtrcl5  37444  dfrcl2  37474  relexpiidm  37504  uspgrsprfo  41065  rngcidALTV  41300  ringcidALTV  41363
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