Theorem fnimaeq0 5485

Description: Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Assertion

Ref	Expression
fnimaeq0	|- ( ( F Fn A /\ B C_ A ) -> ( ( F " B ) = (/) <-> B = (/) ) )

Proof of Theorem fnimaeq0

Step	Hyp	Ref	Expression
1		imadisj 5129	. 2 |- ( ( F " B ) = (/) <-> ( dom F i^i B ) = (/) )
2		incom 3415	. . . 4 |- ( dom F i^i B ) = ( B i^i dom F )
3		fndm 5460	. . . . . . 7 |- ( F Fn A -> dom F = A )
4	3	sseq2d 3272	. . . . . 6 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
5	4	biimpar 297	. . . . 5 |- ( ( F Fn A /\ B C_ A ) -> B C_ dom F )
6		df-ss 3227	. . . . 5 |- ( B C_ dom F <-> ( B i^i dom F ) = B )
7	5, 6	sylib 122	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( B i^i dom F ) = B )
8	2, 7	eqtrid 2279	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( dom F i^i B ) = B )
9	8	eqeq1d 2243	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( ( dom F i^i B ) = (/) <-> B = (/) ) )
10	1, 9	bitrid 192	1 |- ( ( F Fn A /\ B C_ A ) -> ( ( F " B ) = (/) <-> B = (/) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   i^i cin 3213   C_ wss 3214   (/)c0 3512   dom cdm 4754   "cima 4757   Fn wfn 5352

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-fn 5360

This theorem is referenced by: (None)