Theorem resiexg 5103

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 5086	. . 3 |- Rel ( _I |` A )
2		simpr 110	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2301	. . . . . 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 2824	. . . . . 6 |- y e. _V
7	6	opelres 5063	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 4126	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 4927	. . . . . . 7 |- ( x _I y <-> x = y )
10	8, 9	bitr3i 186	. . . . . 6 |- ( <. x , y >. e. _I <-> x = y )
11	10	anbi1i 462	. . . . 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 4799	. . . 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 4861	. 2 |- ( _I |` A ) C_ ( A X. A )
16		xpexg 4884	. . 3 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
17	16	anidms 401	. 2 |- ( A e. V -> ( A X. A ) e. _V )
18		ssexg 4267	. 2 |- ( ( ( _I |` A ) C_ ( A X. A ) /\ ( A X. A ) e. _V ) -> ( _I |` A ) e. _V )
19	15, 17, 18	sylancr 418	1 |- ( A e. V -> ( _I |` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   _Vcvv 2821   C_ wss 3220   <.cop 3708   class class class wbr 4125   _I cid 4428   X. cxp 4767   |` cres 4771

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-res 4781

This theorem is referenced by:  ordiso  7366  omct  7447  ctssexmid  7480  ssomct  13314  ndxarg  13353  ausgrusgrben  16323  subctctexmid  16944