Theorem resiexg 4928

Description: The existence of a restricted identity function, proved without using the Axiom of Replacement. (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 4911	. . 3 |- Rel ( _I |` A )
2		simpr 109	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2228	. . . . . 6 |- ( x = y -> ( x e. A <-> y e. A ) )
4	3	biimpa 294	. . . . 5 |- ( ( x = y /\ x e. A ) -> y e. A )
5	2, 4	jca 304	. . . 4 |- ( ( x = y /\ x e. A ) -> ( x e. A /\ y e. A ) )
6		vex 2728	. . . . . 6 |- y e. _V
7	6	opelres 4888	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 3982	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 4755	. . . . . . 7 |- ( x _I y <-> x = y )
10	8, 9	bitr3i 185	. . . . . 6 |- ( <. x , y >. e. _I <-> x = y )
11	10	anbi1i 454	. . . . 5 |- ( ( <. x , y >. e. _I /\ x e. A ) <-> ( x = y /\ x e. A ) )
12	7, 11	bitri 183	. . . 4 |- ( <. x , y >. e. ( _I |` A ) <-> ( x = y /\ x e. A ) )
13		opelxp 4633	. . . 4 |- ( <. x , y >. e. ( A X. A ) <-> ( x e. A /\ y e. A ) )
14	5, 12, 13	3imtr4i 200	. . 3 |- ( <. x , y >. e. ( _I |` A ) -> <. x , y >. e. ( A X. A ) )
15	1, 14	relssi 4694	. 2 |- ( _I |` A ) C_ ( A X. A )
16		xpexg 4717	. . 3 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
17	16	anidms 395	. 2 |- ( A e. V -> ( A X. A ) e. _V )
18		ssexg 4120	. 2 |- ( ( ( _I |` A ) C_ ( A X. A ) /\ ( A X. A ) e. _V ) -> ( _I |` A ) e. _V )
19	15, 17, 18	sylancr 411	1 |- ( A e. V -> ( _I |` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   _Vcvv 2725   C_ wss 3115   <.cop 3578   class class class wbr 3981   _I cid 4265   X. cxp 4601   |` cres 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186  ax-un 4410

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-br 3982  df-opab 4043  df-id 4270  df-xp 4609  df-rel 4610  df-res 4615

This theorem is referenced by:  ordiso  6997  omct  7078  ctssexmid  7110  ssomct  12374  ndxarg  12413  subctctexmid  13841