Theorem fimacnv 5296

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 4679	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4527	. . . 4 |- dom F = ran `' F
3		fdm 5050	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 2964	. . . . 5 |- A C_ A
5	3, 4	syl6eqss 2995	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	syl5eqssr 2990	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	syl5ss 2956	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 4679	. . . 4 |- ( F " A ) C_ ran F
9		frn 5052	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	syl5ss 2956	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5048	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	syl5sseqr 2994	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5283	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
14	11, 12, 13	syl2anc 391	. . 3 |- ( F : A --> B -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
15	10, 14	mpbid 135	. 2 |- ( F : A --> B -> A C_ ( `' F " B ) )
16	7, 15	eqssd 2962	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   C_ wss 2917   `'ccnv 4344   dom cdm 4345   ran crn 4346   "cima 4348   Fun wfun 4896   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910

This theorem is referenced by:  fmpt  5319  nn0supp  8234