Theorem imai 5099

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 5085	. 2 |- ( _I " A ) = { y | E. x ( x e. A /\ <. x , y >. e. _I ) }
2		df-br 4094	. . . . . . . 8 |- ( x _I y <-> <. x , y >. e. _I )
3		vex 2806	. . . . . . . . 9 |- y e. _V
4	3	ideq 4888	. . . . . . . 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 1654	. . . 4 |- ( E. x ( x e. A /\ <. x , y >. e. _I ) <-> E. x ( x = y /\ x e. A ) )
10		eleq1 2294	. . . . 5 |- ( x = y -> ( x e. A <-> y e. A ) )
11	3, 10	ceqsexv 2843	. . . 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 2347	. 2 |- { y | E. x ( x e. A /\ <. x , y >. e. _I ) } = { y | y e. A }
14		abid2 2353	. 2 |- { y | y e. A } = A
15	1, 13, 14	3eqtri 2256	1 |- ( _I " A ) = A

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2202   {cab 2217   <.cop 3676   class class class wbr 4093   _I cid 4391   "cima 4734

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-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  rnresi  5100  cnvresid  5411  ecidsn  6794