Theorem resiexg 7102

Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 6479). (Contributed by NM, 13-Jan-2007.)

Assertion

Ref	Expression
resiexg	|- ( A e. V -> ( _I |` A ) e. _V )

Proof of Theorem resiexg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 5426	. . 3 |- Rel ( _I |` A )
2		simpr 477	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2689	. . . . . 6 |- ( x = y -> ( x e. A <-> y e. A ) )
4	3	biimpa 501	. . . . 5 |- ( ( x = y /\ x e. A ) -> y e. A )
5	2, 4	jca 554	. . . 4 |- ( ( x = y /\ x e. A ) -> ( x e. A /\ y e. A ) )
6		vex 3203	. . . . . 6 |- y e. _V
7	6	opelres 5401	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 4654	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 5274	. . . . . . 7 |- ( x _I y <-> x = y )
10	8, 9	bitr3i 266	. . . . . 6 |- ( <. x , y >. e. _I <-> x = y )
11	10	anbi1i 731	. . . . 5 |- ( ( <. x , y >. e. _I /\ x e. A ) <-> ( x = y /\ x e. A ) )
12	7, 11	bitri 264	. . . 4 |- ( <. x , y >. e. ( _I |` A ) <-> ( x = y /\ x e. A ) )
13		opelxp 5146	. . . 4 |- ( <. x , y >. e. ( A X. A ) <-> ( x e. A /\ y e. A ) )
14	5, 12, 13	3imtr4i 281	. . 3 |- ( <. x , y >. e. ( _I |` A ) -> <. x , y >. e. ( A X. A ) )
15	1, 14	relssi 5211	. 2 |- ( _I |` A ) C_ ( A X. A )
16		sqxpexg 6963	. 2 |- ( A e. V -> ( A X. A ) e. _V )
17		ssexg 4804	. 2 |- ( ( ( _I |` A ) C_ ( A X. A ) /\ ( A X. A ) e. _V ) -> ( _I |` A ) e. _V )
18	15, 16, 17	sylancr 695	1 |- ( A e. V -> ( _I |` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   <.cop 4183   class class class wbr 4653   _I cid 5023   X. cxp 5112   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  ordiso  8421  wdomref  8477  dfac9  8958  relexp0g  13762  relexpsucnnr  13765  ndxarg  15882  idfu2nd  16537  idfu1st  16539  idfucl  16541  setcid  16736  equivestrcsetc  16792  pf1ind  19719  islinds2  20152  ausgrusgrb  26060  upgrres1lem1  26201  cusgrexilem1  26335  sizusglecusg  26359  pliguhgr  27338  bj-evalid  33028  poimirlem15  33424  dib0  36453  dicn0  36481  cdlemn11a  36496  dihord6apre  36545  dihatlat  36623  dihpN  36625  eldioph2lem1  37323  eldioph2lem2  37324  dfrtrcl5  37936  dfrcl2  37966  relexpiidm  37996  uspgrsprfo  41756  rngcidALTV  41991  ringcidALTV  42054