Theorem fimacnvdisj 5372

Description: The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)

Assertion

Ref	Expression
fimacnvdisj	|- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ( `' F " C ) = (/) )

Proof of Theorem fimacnvdisj

Step	Hyp	Ref	Expression
1		df-rn 4615	. . . 4 |- ran F = dom `' F
2		frn 5346	. . . . 5 |- ( F : A --> B -> ran F C_ B )
3	2	adantr 274	. . . 4 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ran F C_ B )
4	1, 3	eqsstrrid 3189	. . 3 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> dom `' F C_ B )
5		ssdisj 3465	. . 3 |- ( ( dom `' F C_ B /\ ( B i^i C ) = (/) ) -> ( dom `' F i^i C ) = (/) )
6	4, 5	sylancom 417	. 2 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ( dom `' F i^i C ) = (/) )
7		imadisj 4966	. 2 |- ( ( `' F " C ) = (/) <-> ( dom `' F i^i C ) = (/) )
8	6, 7	sylibr 133	1 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ( `' F " C ) = (/) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   i^i cin 3115   C_ wss 3116   (/)c0 3409   `'ccnv 4603   dom cdm 4604   ran crn 4605   "cima 4607   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-f 5192

This theorem is referenced by: (None)