Theorem fimacnv 5811

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 5117	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4953	. . . 4 |- dom F = ran `' F
3		fdm 5519	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3262	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3294	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3289	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3253	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 5117	. . . 4 |- ( F " A ) C_ ran F
9		frn 5522	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3253	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5516	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3293	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5799	. . . 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 3259	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   C_ wss 3214   `'ccnv 4753   dom cdm 4754   ran crn 4755   "cima 4757   Fun wfun 5351   -->wf 5353

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365

This theorem is referenced by:  fmpt  5832  fsuppeq  6460  fsuppeqg  6461  nn0supp  9569  cnclima  15214