Theorem fnimadisj 5396

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 5373	. . . . 5 |- ( F Fn A -> dom F = A )
2	1	ineq1d 3373	. . . 4 |- ( F Fn A -> ( dom F i^i C ) = ( A i^i C ) )
3	2	eqeq1d 2214	. . 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 5044	. 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 1373   i^i cin 3165   (/)c0 3460   dom cdm 4675   "cima 4678   Fn wfn 5266

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-fn 5274

This theorem is referenced by: (None)