Theorem elpreima 10161

Description: Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
elpreima	|- (F Fn A -> (B e. (`'F"C) <-> (B e. A /\ (F` B) e. C)))

Proof of Theorem elpreima

Step	Hyp	Ref	Expression
1		fndm 4512	. . . . 5 |- (F Fn A -> dom F = A)
2	1	eleq2d 1964	. . . 4 |- (F Fn A -> (B e. dom F <-> B e. A))
3		cnvimass 4286	. . . . 5 |- (`'F"C) C_ dom F
4	3	sseli 2617	. . . 4 |- (B e. (`'F"C) -> B e. dom F)
5	2, 4	syl5bi 225	. . 3 |- (F Fn A -> (B e. (`'F"C) -> B e. A))
6		fvimacnvi 4777	. . . . 5 |- ((Fun F /\ B e. (`'F"C)) -> (F` B) e. C)
7		fnfun 4510	. . . . 5 |- (F Fn A -> Fun F)
8	6, 7	sylan 497	. . . 4 |- ((F Fn A /\ B e. (`'F"C)) -> (F` B) e. C)
9	8	ex 402	. . 3 |- (F Fn A -> (B e. (`'F"C) -> (F` B) e. C))
10	5, 9	jcad 661	. 2 |- (F Fn A -> (B e. (`'F"C) -> (B e. A /\ (F` B) e. C)))
11		fvimacnv 4778	. . . . 5 |- ((Fun F /\ B e. dom F) -> ((F` B) e. C <-> B e. (`'F"C)))
12	11	funfni 4513	. . . 4 |- ((F Fn A /\ B e. A) -> ((F` B) e. C <-> B e. (`'F"C)))
13	12	biimpd 170	. . 3 |- ((F Fn A /\ B e. A) -> ((F` B) e. C -> B e. (`'F"C)))
14	13	expimpd 404	. 2 |- (F Fn A -> ((B e. A /\ (F` B) e. C) -> B e. (`'F"C)))
15	10, 14	impbid 574	1 |- (F Fn A -> (B e. (`'F"C) <-> (B e. A /\ (F` B) e. C)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  `'ccnv 3985  dom cdm 3986  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  uptx 10226  inpreima 15682  unpreima 15683  respreima 15684  cnresima 15891  keridl 16180

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

Copyright terms: Public domain