Theorem fnimaeq0 5121

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 4781	. 2 |- ( ( F " B ) = (/) <-> ( dom F i^i B ) = (/) )
2		incom 3190	. . . 4 |- ( dom F i^i B ) = ( B i^i dom F )
3		fndm 5099	. . . . . . 7 |- ( F Fn A -> dom F = A )
4	3	sseq2d 3052	. . . . . 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 3010	. . . . 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 2132	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( dom F i^i B ) = B )
9	8	eqeq1d 2096	. 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 1289   i^i cin 2996   C_ wss 2997   (/)c0 3284   dom cdm 4428   "cima 4431   Fn wfn 4997

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-fn 5005

This theorem is referenced by: (None)