Theorem fimacnv 5776

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 5087	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4923	. . . 4 |- dom F = ran `' F
3		fdm 5488	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3247	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3279	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3274	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3238	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 5087	. . . 4 |- ( F " A ) C_ ran F
9		frn 5491	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3238	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5485	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3278	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5763	. . . 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 3244	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   C_ wss 3200   `'ccnv 4724   dom cdm 4725   ran crn 4726   "cima 4728   Fun wfun 5320   -->wf 5322

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by:  fmpt  5797  nn0supp  9453  cnclima  14946