Theorem fimacnv 5428

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 4785	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4628	. . . 4 |- dom F = ran `' F
3		fdm 5166	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3044	. . . . 5 |- A C_ A
5	3, 4	syl6eqss 3076	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	syl5eqssr 3071	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	syl5ss 3036	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 4785	. . . 4 |- ( F " A ) C_ ran F
9		frn 5169	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	syl5ss 3036	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5164	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	syl5sseqr 3075	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5415	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
14	11, 12, 13	syl2anc 403	. . 3 |- ( F : A --> B -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
15	10, 14	mpbid 145	. 2 |- ( F : A --> B -> A C_ ( `' F " B ) )
16	7, 15	eqssd 3042	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   C_ wss 2999   `'ccnv 4437   dom cdm 4438   ran crn 4439   "cima 4441   Fun wfun 5009   -->wf 5011

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023

This theorem is referenced by:  fmpt  5449  nn0supp  8723