Theorem fimacnv 5593

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 4936	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4775	. . . 4 |- dom F = ran `' F
3		fdm 5322	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3148	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3180	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3175	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3139	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 4936	. . . 4 |- ( F " A ) C_ ran F
9		frn 5325	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3139	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5319	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3179	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5580	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
14	11, 12, 13	syl2anc 409	. . 3 |- ( F : A --> B -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
15	10, 14	mpbid 146	. 2 |- ( F : A --> B -> A C_ ( `' F " B ) )
16	7, 15	eqssd 3145	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1335   C_ wss 3102   `'ccnv 4582   dom cdm 4583   ran crn 4584   "cima 4586   Fun wfun 5161   -->wf 5163

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175

This theorem is referenced by:  fmpt  5614  nn0supp  9125  cnclima  12583