Theorem fnimadisj 5479

Description: A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)

Assertion

Ref	Expression
fnimadisj	|- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( F " C ) = (/) )

Proof of Theorem fnimadisj

Step	Hyp	Ref	Expression
1		fndm 5455	. . . . 5 |- ( F Fn A -> dom F = A )
2	1	ineq1d 3421	. . . 4 |- ( F Fn A -> ( dom F i^i C ) = ( A i^i C ) )
3	2	eqeq1d 2241	. . 3 |- ( F Fn A -> ( ( dom F i^i C ) = (/) <-> ( A i^i C ) = (/) ) )
4	3	biimpar 297	. 2 |- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( dom F i^i C ) = (/) )
5		imadisj 5124	. 2 |- ( ( F " C ) = (/) <-> ( dom F i^i C ) = (/) )
6	4, 5	sylibr 134	1 |- ( ( F Fn A /\ ( A i^i C ) = (/) ) -> ( F " C ) = (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   i^i cin 3210   (/)c0 3508   dom cdm 4749   "cima 4752   Fn wfn 5347

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-fn 5355

This theorem is referenced by: (None)