Theorem resiexg 4859

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 4842	. . 3 |- Rel ( _I |` A )
2		simpr 109	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2200	. . . . . 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 2684	. . . . . 6 |- y e. _V
7	6	opelres 4819	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 3925	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 4686	. . . . . . 7 |- ( x _I y <-> x = y )
10	8, 9	bitr3i 185	. . . . . 6 |- ( <. x , y >. e. _I <-> x = y )
11	10	anbi1i 453	. . . . 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 4564	. . . 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 4625	. 2 |- ( _I |` A ) C_ ( A X. A )
16		xpexg 4648	. . 3 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
17	16	anidms 394	. 2 |- ( A e. V -> ( A X. A ) e. _V )
18		ssexg 4062	. 2 |- ( ( ( _I |` A ) C_ ( A X. A ) /\ ( A X. A ) e. _V ) -> ( _I |` A ) e. _V )
19	15, 17, 18	sylancr 410	1 |- ( A e. V -> ( _I |` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   _Vcvv 2681   C_ wss 3066   <.cop 3525   class class class wbr 3924   _I cid 4205   X. cxp 4532   |` cres 4536

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-res 4546

This theorem is referenced by:  ordiso  6914  omct  6995  ctssexmid  7017  ndxarg  11971  subctctexmid  13185