Theorem fimacnv 5645

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 4981	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4819	. . . 4 |- dom F = ran `' F
3		fdm 5371	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3175	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3207	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3202	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3166	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 4981	. . . 4 |- ( F " A ) C_ ran F
9		frn 5374	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3166	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5368	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3206	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5632	. . . 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 3172	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   C_ wss 3129   `'ccnv 4625   dom cdm 4626   ran crn 4627   "cima 4629   Fun wfun 5210   -->wf 5212

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224

This theorem is referenced by:  fmpt  5666  nn0supp  9226  cnclima  13616