Theorem resiexg 4910

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 4893	. . 3 |- Rel ( _I |` A )
2		simpr 109	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2220	. . . . . 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 2715	. . . . . 6 |- y e. _V
7	6	opelres 4870	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 3966	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 4737	. . . . . . 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 4615	. . . 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 4676	. 2 |- ( _I |` A ) C_ ( A X. A )
16		xpexg 4699	. . 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 4103	. 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 2128   _Vcvv 2712   C_ wss 3102   <.cop 3563   class class class wbr 3965   _I cid 4248   X. cxp 4583   |` cres 4587

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-res 4597

This theorem is referenced by:  ordiso  6975  omct  7056  ctssexmid  7088  ssomct  12161  ndxarg  12200  subctctexmid  13560