Theorem resiexg 4872

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 4855	. . 3 |- Rel ( _I |` A )
2		simpr 109	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2203	. . . . . 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 2692	. . . . . 6 |- y e. _V
7	6	opelres 4832	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 3938	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 4699	. . . . . . 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 4577	. . . 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 4638	. 2 |- ( _I |` A ) C_ ( A X. A )
16		xpexg 4661	. . 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 4075	. 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 1481   _Vcvv 2689   C_ wss 3076   <.cop 3535   class class class wbr 3937   _I cid 4218   X. cxp 4545   |` cres 4549

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-res 4559

This theorem is referenced by:  ordiso  6929  omct  7010  ctssexmid  7032  ndxarg  12021  subctctexmid  13369