Theorem fimacnv 5764

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 5079	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4915	. . . 4 |- dom F = ran `' F
3		fdm 5479	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3244	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3276	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3271	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3235	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 5079	. . . 4 |- ( F " A ) C_ ran F
9		frn 5482	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3235	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5476	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3275	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5751	. . . 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 3241	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   C_ wss 3197   `'ccnv 4718   dom cdm 4719   ran crn 4720   "cima 4722   Fun wfun 5312   -->wf 5314

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326

This theorem is referenced by:  fmpt  5785  nn0supp  9421  cnclima  14897