Theorem resiexg 4936

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 4919	. . 3 |- Rel ( _I |` A )
2		simpr 109	. . . . 5 |- ( ( x = y /\ x e. A ) -> x e. A )
3		eleq1 2233	. . . . . 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 2733	. . . . . 6 |- y e. _V
7	6	opelres 4896	. . . . 5 |- ( <. x , y >. e. ( _I |` A ) <-> ( <. x , y >. e. _I /\ x e. A ) )
8		df-br 3990	. . . . . . 7 |- ( x _I y <-> <. x , y >. e. _I )
9	6	ideq 4763	. . . . . . 7 |- ( x _I y <-> x = y )
10	8, 9	bitr3i 185	. . . . . 6 |- ( <. x , y >. e. _I <-> x = y )
11	10	anbi1i 455	. . . . 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 4641	. . . 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 4702	. 2 |- ( _I |` A ) C_ ( A X. A )
16		xpexg 4725	. . 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 4128	. 2 |- ( ( ( _I |` A ) C_ ( A X. A ) /\ ( A X. A ) e. _V ) -> ( _I |` A ) e. _V )
19	15, 17, 18	sylancr 412	1 |- ( A e. V -> ( _I |` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   _Vcvv 2730   C_ wss 3121   <.cop 3586   class class class wbr 3989   _I cid 4273   X. cxp 4609   |` cres 4613

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-res 4623

This theorem is referenced by:  ordiso  7013  omct  7094  ctssexmid  7126  ssomct  12400  ndxarg  12439  subctctexmid  14034