Theorem fimacnvdisj 5551

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 4760	. . . 4 |- ran F = dom `' F
2		frn 5517	. . . . 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 3285	. . 3 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> dom `' F C_ B )
5		ssdisj 3565	. . 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 5124	. 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 1398   i^i cin 3210   C_ wss 3211   (/)c0 3508   `'ccnv 4748   dom cdm 4749   ran crn 4750   "cima 4752   -->wf 5348

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-f 5356

This theorem is referenced by: (None)