Theorem imai 4853

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 4842	. 2 |- ( _I " A ) = { y | E. x ( x e. A /\ <. x , y >. e. _I ) }
2		df-br 3896	. . . . . . . 8 |- ( x _I y <-> <. x , y >. e. _I )
3		vex 2660	. . . . . . . . 9 |- y e. _V
4	3	ideq 4651	. . . . . . . 8 |- ( x _I y <-> x = y )
5	2, 4	bitr3i 185	. . . . . . 7 |- ( <. x , y >. e. _I <-> x = y )
6	5	anbi2i 450	. . . . . 6 |- ( ( x e. A /\ <. x , y >. e. _I ) <-> ( x e. A /\ x = y ) )
7		ancom 264	. . . . . 6 |- ( ( x e. A /\ x = y ) <-> ( x = y /\ x e. A ) )
8	6, 7	bitri 183	. . . . 5 |- ( ( x e. A /\ <. x , y >. e. _I ) <-> ( x = y /\ x e. A ) )
9	8	exbii 1567	. . . 4 |- ( E. x ( x e. A /\ <. x , y >. e. _I ) <-> E. x ( x = y /\ x e. A ) )
10		eleq1 2177	. . . . 5 |- ( x = y -> ( x e. A <-> y e. A ) )
11	3, 10	ceqsexv 2696	. . . 4 |- ( E. x ( x = y /\ x e. A ) <-> y e. A )
12	9, 11	bitri 183	. . 3 |- ( E. x ( x e. A /\ <. x , y >. e. _I ) <-> y e. A )
13	12	abbii 2230	. 2 |- { y | E. x ( x e. A /\ <. x , y >. e. _I ) } = { y | y e. A }
14		abid2 2235	. 2 |- { y | y e. A } = A
15	1, 13, 14	3eqtri 2139	1 |- ( _I " A ) = A

Syntax hints:   /\ wa 103   = wceq 1314   E.wex 1451   e. wcel 1463   {cab 2101   <.cop 3496   class class class wbr 3895   _I cid 4170   "cima 4502

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512

This theorem is referenced by:  rnresi  4854  cnvresid  5155  ecidsn  6430