Theorem fimacnv 5694

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 5021	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4859	. . . 4 |- dom F = ran `' F
3		fdm 5416	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3204	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3236	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3231	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3195	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 5021	. . . 4 |- ( F " A ) C_ ran F
9		frn 5419	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3195	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5413	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3235	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5681	. . . 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 3201	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   C_ wss 3157   `'ccnv 4663   dom cdm 4664   ran crn 4665   "cima 4667   Fun wfun 5253   -->wf 5255

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267

This theorem is referenced by:  fmpt  5715  nn0supp  9318  cnclima  14543