Theorem fimacnv 5806

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 5112	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4948	. . . 4 |- dom F = ran `' F
3		fdm 5514	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3258	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3290	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3285	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3249	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 5112	. . . 4 |- ( F " A ) C_ ran F
9		frn 5517	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3249	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5511	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3289	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5794	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
14	11, 12, 13	syl2anc 411	. . 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 3255	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   C_ wss 3211   `'ccnv 4748   dom cdm 4749   ran crn 4750   "cima 4752   Fun wfun 5346   -->wf 5348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360

This theorem is referenced by:  fmpt  5827  fsuppeq  6447  fsuppeqg  6448  nn0supp  9552  cnclima  15088