Theorem resiexg 5082

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 5065	. . 3 |- Rel ( _I |` A )
2		simpr 110	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2295	. . . . . 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 2815	. . . . . 6 |- y e. _V
7	6	opelres 5042	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 4109	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 4906	. . . . . . 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 4778	. . . 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 4840	. 2 |- ( _I |` A ) C_ ( A X. A )
16		xpexg 4863	. . 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 4248	. 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 2203   _Vcvv 2812   C_ wss 3210   <.cop 3691   class class class wbr 4108   _I cid 4408   X. cxp 4746   |` cres 4750

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-res 4760

This theorem is referenced by:  ordiso  7326  omct  7407  ctssexmid  7440  ssomct  13188  ndxarg  13227  ausgrusgrben  16155  subctctexmid  16766