Theorem fimacnv 5837

Description: The preimage 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 5137	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4973	. . . 4 |- dom F = ran `' F
3		fdm 5539	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3268	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3300	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3295	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3259	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 5137	. . . 4 |- ( F " A ) C_ ran F
9		frn 5542	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3259	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5536	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3299	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5825	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
14	11, 12, 13	syl2anc 415	. . 3 |- ( F : A --> B -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
15	10, 14	mpbid 147	. 2 |- ( F : A --> B -> A C_ ( `' F " B ) )
16	7, 15	eqssd 3265	1 |- ( F : A --> B -> ( `' F " B ) = A )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   C_ wss 3220   `'ccnv 4773   dom cdm 4774   ran crn 4775   "cima 4777   Fun wfun 5371   -->wf 5373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385

This theorem is used by:  fmpt  5858  fsuppeq  6487  fsuppeqg  6488  nn0supp  9619  cnclima  15324