Theorem imai 5478

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 5469	. 2 |- ( _I " A ) = { y | E. x ( x e. A /\ <. x , y >. e. _I ) }
2		df-br 4654	. . . . . . . 8 |- ( x _I y <-> <. x , y >. e. _I )
3		vex 3203	. . . . . . . . 9 |- y e. _V
4	3	ideq 5274	. . . . . . . 8 |- ( x _I y <-> x = y )
5	2, 4	bitr3i 266	. . . . . . 7 |- ( <. x , y >. e. _I <-> x = y )
6	5	anbi2i 730	. . . . . 6 |- ( ( x e. A /\ <. x , y >. e. _I ) <-> ( x e. A /\ x = y ) )
7		ancom 466	. . . . . 6 |- ( ( x e. A /\ x = y ) <-> ( x = y /\ x e. A ) )
8	6, 7	bitri 264	. . . . 5 |- ( ( x e. A /\ <. x , y >. e. _I ) <-> ( x = y /\ x e. A ) )
9	8	exbii 1774	. . . 4 |- ( E. x ( x e. A /\ <. x , y >. e. _I ) <-> E. x ( x = y /\ x e. A ) )
10		eleq1 2689	. . . . 5 |- ( x = y -> ( x e. A <-> y e. A ) )
11	10	equsexvw 1932	. . . 4 |- ( E. x ( x = y /\ x e. A ) <-> y e. A )
12	9, 11	bitri 264	. . 3 |- ( E. x ( x e. A /\ <. x , y >. e. _I ) <-> y e. A )
13	12	abbii 2739	. 2 |- { y | E. x ( x e. A /\ <. x , y >. e. _I ) } = { y | y e. A }
14		abid2 2745	. 2 |- { y | y e. A } = A
15	1, 13, 14	3eqtri 2648	1 |- ( _I " A ) = A

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   <.cop 4183   class class class wbr 4653   _I cid 5023   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  rnresi  5479  cnvresid  5968  ecidsn  7795  mbfid  23403  frege131d  38056  frege110  38267  frege133  38290