Theorem fnimadisj 5348

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 5327	. . . . 5 |- ( F Fn A -> dom F = A )
2	1	ineq1d 3347	. . . 4 |- ( F Fn A -> ( dom F i^i C ) = ( A i^i C ) )
3	2	eqeq1d 2196	. . 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 5002	. 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 1363   i^i cin 3140   (/)c0 3434   dom cdm 4638   "cima 4641   Fn wfn 5223

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-fn 5231

This theorem is referenced by: (None)