Theorem fimacnv 4783

Description: The pre-image of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

Assertion

Ref	Expression
fimacnv	|- (F:A-->B -> (`'F"B) = A)

Proof of Theorem fimacnv

Step	Hyp	Ref	Expression
1		imassrn 4278	. . . 4 |- (`'F"B) C_ ran `' F
2	1	a1i 8	. . 3 |- (F:A-->B -> (`'F"B) C_ ran `' F)
3		fdm 4567	. . . . 5 |- (F:A-->B -> dom F = A)
4		ssid 2634	. . . . . 6 |- A C_ A
5	4	a1i 8	. . . . 5 |- (F:A-->B -> A C_ A)
6	3, 5	eqsstrd 2651	. . . 4 |- (F:A-->B -> dom F C_ A)
7		dfdm4 4151	. . . 4 |- dom F = ran `' F
8	6, 7	syl5ssr 2662	. . 3 |- (F:A-->B -> ran `' F C_ A)
9	2, 8	sstrd 2627	. 2 |- (F:A-->B -> (`'F"B) C_ A)
10		imassrn 4278	. . . . 5 |- (F"A) C_ ran F
11	10	a1i 8	. . . 4 |- (F:A-->B -> (F"A) C_ ran F)
12		frn 4569	. . . 4 |- (F:A-->B -> ran F C_ B)
13	11, 12	sstrd 2627	. . 3 |- (F:A-->B -> (F"A) C_ B)
14		ffun 4565	. . . 4 |- (F:A-->B -> Fun F)
15	5, 3	sseqtr4d 2654	. . . 4 |- (F:A-->B -> A C_ dom F)
16		funimass3 4779	. . . 4 |- ((Fun F /\ A C_ dom F) -> ((F"A) C_ B <-> A C_ (`'F"B)))
17	14, 15, 16	syl11anc 524	. . 3 |- (F:A-->B -> ((F"A) C_ B <-> A C_ (`'F"B)))
18	13, 17	mpbid 212	. 2 |- (F:A-->B -> A C_ (`'F"B))
19	9, 18	eqssd 2633	1 |- (F:A-->B -> (`'F"B) = A)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   C_ wss 2593  `'ccnv 3985  dom cdm 3986  ran crn 3987  "cima 3989  Fun wfun 3992  -->wf 3994

This theorem is referenced by:  iscncl 9047  mapudiscn 14872  eqindhome 14895  cnpfillim 15589  rnelfmlem 15592  rnelfm 15593  fmfnfm 15598  cnss 15892

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-ral 2109  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-f 4010  df-fv 4014

Copyright terms: Public domain