Theorem fimacnvdisj 5445

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 4675	. . . 4 |- ran F = dom `' F
2		frn 5419	. . . . 5 |- ( F : A --> B -> ran F C_ B )
3	2	adantr 276	. . . 4 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ran F C_ B )
4	1, 3	eqsstrrid 3231	. . 3 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> dom `' F C_ B )
5		ssdisj 3508	. . 3 |- ( ( dom `' F C_ B /\ ( B i^i C ) = (/) ) -> ( dom `' F i^i C ) = (/) )
6	4, 5	sylancom 420	. 2 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ( dom `' F i^i C ) = (/) )
7		imadisj 5032	. 2 |- ( ( `' F " C ) = (/) <-> ( dom `' F i^i C ) = (/) )
8	6, 7	sylibr 134	1 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ( `' F " C ) = (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   i^i cin 3156   C_ wss 3157   (/)c0 3451   `'ccnv 4663   dom cdm 4664   ran crn 4665   "cima 4667   -->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-in1 615  ax-in2 616  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-fal 1370  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-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-f 5263

This theorem is referenced by: (None)