Theorem fimacnv 5614

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 4957	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4796	. . . 4 |- dom F = ran `' F
3		fdm 5343	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3162	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3194	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3189	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3153	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 4957	. . . 4 |- ( F " A ) C_ ran F
9		frn 5346	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3153	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5340	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3193	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5601	. . . 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 3159	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   C_ wss 3116   `'ccnv 4603   dom cdm 4604   ran crn 4605   "cima 4607   Fun wfun 5182   -->wf 5184

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196

This theorem is referenced by:  fmpt  5635  nn0supp  9166  cnclima  12863