Theorem resiexg 4953

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 4936	. . 3 |- Rel ( _I |` A )
2		simpr 110	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2240	. . . . . 6 |- ( x = y -> ( x e. A <-> y e. A ) )
4	3	biimpa 296	. . . . 5 |- ( ( x = y /\ x e. A ) -> y e. A )
5	2, 4	jca 306	. . . 4 |- ( ( x = y /\ x e. A ) -> ( x e. A /\ y e. A ) )
6		vex 2741	. . . . . 6 |- y e. _V
7	6	opelres 4913	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 4005	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 4780	. . . . . . 7 |- ( x _I y <-> x = y )
10	8, 9	bitr3i 186	. . . . . 6 |- ( <. x , y >. e. _I <-> x = y )
11	10	anbi1i 458	. . . . 5 |- ( ( <. x , y >. e. _I /\ x e. A ) <-> ( x = y /\ x e. A ) )
12	7, 11	bitri 184	. . . 4 |- ( <. x , y >. e. ( _I |` A ) <-> ( x = y /\ x e. A ) )
13		opelxp 4657	. . . 4 |- ( <. x , y >. e. ( A X. A ) <-> ( x e. A /\ y e. A ) )
14	5, 12, 13	3imtr4i 201	. . 3 |- ( <. x , y >. e. ( _I |` A ) -> <. x , y >. e. ( A X. A ) )
15	1, 14	relssi 4718	. 2 |- ( _I |` A ) C_ ( A X. A )
16		xpexg 4741	. . 3 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
17	16	anidms 397	. 2 |- ( A e. V -> ( A X. A ) e. _V )
18		ssexg 4143	. 2 |- ( ( ( _I |` A ) C_ ( A X. A ) /\ ( A X. A ) e. _V ) -> ( _I |` A ) e. _V )
19	15, 17, 18	sylancr 414	1 |- ( A e. V -> ( _I |` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   _Vcvv 2738   C_ wss 3130   <.cop 3596   class class class wbr 4004   _I cid 4289   X. cxp 4625   |` cres 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-res 4639

This theorem is referenced by:  ordiso  7035  omct  7116  ctssexmid  7148  ssomct  12446  ndxarg  12485  subctctexmid  14753