Theorem fnimaeq0 5040

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 4707	. 2 |- ( ( F " B ) = (/) <-> ( dom F i^i B ) = (/) )
2		incom 3158	. . . 4 |- ( dom F i^i B ) = ( B i^i dom F )
3		fndm 5018	. . . . . . 7 |- ( F Fn A -> dom F = A )
4	3	sseq2d 3027	. . . . . 6 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
5	4	biimpar 291	. . . . 5 |- ( ( F Fn A /\ B C_ A ) -> B C_ dom F )
6		df-ss 2986	. . . . 5 |- ( B C_ dom F <-> ( B i^i dom F ) = B )
7	5, 6	sylib 120	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( B i^i dom F ) = B )
8	2, 7	syl5eq 2125	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( dom F i^i B ) = B )
9	8	eqeq1d 2089	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( ( dom F i^i B ) = (/) <-> B = (/) ) )
10	1, 9	syl5bb 190	1 |- ( ( F Fn A /\ B C_ A ) -> ( ( F " B ) = (/) <-> B = (/) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   i^i cin 2972   C_ wss 2973   (/)c0 3251   dom cdm 4363   "cima 4366   Fn wfn 4917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-fn 4925

This theorem is referenced by: (None)