Theorem fimacnvdisj 5094

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 4374	. . . 4 |- ran F = dom `' F
2		frn 5072	. . . . 5 |- ( F : A --> B -> ran F C_ B )
3	2	adantr 270	. . . 4 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ran F C_ B )
4	1, 3	syl5eqssr 3044	. . 3 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> dom `' F C_ B )
5		ssdisj 3300	. . 3 |- ( ( dom `' F C_ B /\ ( B i^i C ) = (/) ) -> ( dom `' F i^i C ) = (/) )
6	4, 5	sylancom 411	. 2 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ( dom `' F i^i C ) = (/) )
7		imadisj 4707	. 2 |- ( ( `' F " C ) = (/) <-> ( dom `' F i^i C ) = (/) )
8	6, 7	sylibr 132	1 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> ( `' F " C ) = (/) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   i^i cin 2972   C_ wss 2973   (/)c0 3251   `'ccnv 4362   dom cdm 4363   ran crn 4364   "cima 4366   -->wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-f 4926

This theorem is referenced by: (None)