Theorem imai 4969

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 4958	. 2 |- ( _I " A ) = { y | E. x ( x e. A /\ <. x , y >. e. _I ) }
2		df-br 3991	. . . . . . . 8 |- ( x _I y <-> <. x , y >. e. _I )
3		vex 2734	. . . . . . . . 9 |- y e. _V
4	3	ideq 4764	. . . . . . . 8 |- ( x _I y <-> x = y )
5	2, 4	bitr3i 185	. . . . . . 7 |- ( <. x , y >. e. _I <-> x = y )
6	5	anbi2i 455	. . . . . 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 1599	. . . 4 |- ( E. x ( x e. A /\ <. x , y >. e. _I ) <-> E. x ( x = y /\ x e. A ) )
10		eleq1 2234	. . . . 5 |- ( x = y -> ( x e. A <-> y e. A ) )
11	3, 10	ceqsexv 2770	. . . 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 2287	. 2 |- { y | E. x ( x e. A /\ <. x , y >. e. _I ) } = { y | y e. A }
14		abid2 2292	. 2 |- { y | y e. A } = A
15	1, 13, 14	3eqtri 2196	1 |- ( _I " A ) = A

Syntax hints:   /\ wa 103   = wceq 1349   E.wex 1486   e. wcel 2142   {cab 2157   <.cop 3587   class class class wbr 3990   _I cid 4274   "cima 4615

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-br 3991  df-opab 4052  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625

This theorem is referenced by:  rnresi  4970  cnvresid  5274  ecidsn  6564