Theorem fnimaeq0 5319

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 4973	. 2 |- ( ( F " B ) = (/) <-> ( dom F i^i B ) = (/) )
2		incom 3319	. . . 4 |- ( dom F i^i B ) = ( B i^i dom F )
3		fndm 5297	. . . . . . 7 |- ( F Fn A -> dom F = A )
4	3	sseq2d 3177	. . . . . 6 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
5	4	biimpar 295	. . . . 5 |- ( ( F Fn A /\ B C_ A ) -> B C_ dom F )
6		df-ss 3134	. . . . 5 |- ( B C_ dom F <-> ( B i^i dom F ) = B )
7	5, 6	sylib 121	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( B i^i dom F ) = B )
8	2, 7	eqtrid 2215	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( dom F i^i B ) = B )
9	8	eqeq1d 2179	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( ( dom F i^i B ) = (/) <-> B = (/) ) )
10	1, 9	syl5bb 191	1 |- ( ( F Fn A /\ B C_ A ) -> ( ( F " B ) = (/) <-> B = (/) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   i^i cin 3120   C_ wss 3121   (/)c0 3414   dom cdm 4611   "cima 4614   Fn wfn 5193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fn 5201

This theorem is referenced by: (None)