Theorem fimacnv 5317

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 4699	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4545	. . . 4 |- dom F = ran `' F
3		fdm 5070	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3018	. . . . 5 |- A C_ A
5	3, 4	syl6eqss 3049	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	syl5eqssr 3044	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	syl5ss 3010	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 4699	. . . 4 |- ( F " A ) C_ ran F
9		frn 5072	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	syl5ss 3010	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5068	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	syl5sseqr 3048	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5304	. . . 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 3016	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   C_ wss 2973   `'ccnv 4362   dom cdm 4363   ran crn 4364   "cima 4366   Fun wfun 4916   -->wf 4918

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by:  fmpt  5340  nn0supp  8340