Theorem fimacnv 5688

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 5017	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4855	. . . 4 |- dom F = ran `' F
3		fdm 5410	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3200	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3232	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3227	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3191	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 5017	. . . 4 |- ( F " A ) C_ ran F
9		frn 5413	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3191	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5407	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3231	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5675	. . . 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 3197	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   C_ wss 3154   `'ccnv 4659   dom cdm 4660   ran crn 4661   "cima 4663   Fun wfun 5249   -->wf 5251

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263

This theorem is referenced by:  fmpt  5709  nn0supp  9295  cnclima  14402