Theorem fimacnvdisj 5521

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 4736	. . . 4 |- ran F = dom `' F
2		frn 5491	. . . . 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 3274	. . 3 |- ( ( F : A --> B /\ ( B i^i C ) = (/) ) -> dom `' F C_ B )
5		ssdisj 3551	. . 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 5098	. 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 1397   i^i cin 3199   C_ wss 3200   (/)c0 3494   `'ccnv 4724   dom cdm 4725   ran crn 4726   "cima 4728   -->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-in1 619  ax-in2 620  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-fal 1403  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-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-f 5330

This theorem is referenced by: (None)