Theorem imai 4996

Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Assertion

Ref	Expression
imai	|- ( _I " A ) = A

Proof of Theorem imai

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfima3 4985	. 2 |- ( _I " A ) = { y | E. x ( x e. A /\ <. x , y >. e. _I ) }
2		df-br 4016	. . . . . . . 8 |- ( x _I y <-> <. x , y >. e. _I )
3		vex 2752	. . . . . . . . 9 |- y e. _V
4	3	ideq 4791	. . . . . . . 8 |- ( x _I y <-> x = y )
5	2, 4	bitr3i 186	. . . . . . 7 |- ( <. x , y >. e. _I <-> x = y )
6	5	anbi2i 457	. . . . . 6 |- ( ( x e. A /\ <. x , y >. e. _I ) <-> ( x e. A /\ x = y ) )
7		ancom 266	. . . . . 6 |- ( ( x e. A /\ x = y ) <-> ( x = y /\ x e. A ) )
8	6, 7	bitri 184	. . . . 5 |- ( ( x e. A /\ <. x , y >. e. _I ) <-> ( x = y /\ x e. A ) )
9	8	exbii 1615	. . . 4 |- ( E. x ( x e. A /\ <. x , y >. e. _I ) <-> E. x ( x = y /\ x e. A ) )
10		eleq1 2250	. . . . 5 |- ( x = y -> ( x e. A <-> y e. A ) )
11	3, 10	ceqsexv 2788	. . . 4 |- ( E. x ( x = y /\ x e. A ) <-> y e. A )
12	9, 11	bitri 184	. . 3 |- ( E. x ( x e. A /\ <. x , y >. e. _I ) <-> y e. A )
13	12	abbii 2303	. 2 |- { y | E. x ( x e. A /\ <. x , y >. e. _I ) } = { y | y e. A }
14		abid2 2308	. 2 |- { y | y e. A } = A
15	1, 13, 14	3eqtri 2212	1 |- ( _I " A ) = A

Syntax hints:   /\ wa 104   = wceq 1363   E.wex 1502   e. wcel 2158   {cab 2173   <.cop 3607   class class class wbr 4015   _I cid 4300   "cima 4641

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651

This theorem is referenced by:  rnresi  4997  cnvresid  5302  ecidsn  6595