Theorem fnimaeq0 5020

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 4687	. 2 |- ( ( F " B ) = (/) <-> ( dom F i^i B ) = (/) )
2		incom 3129	. . . 4 |- ( dom F i^i B ) = ( B i^i dom F )
3		fndm 4998	. . . . . . 7 |- ( F Fn A -> dom F = A )
4	3	sseq2d 2973	. . . . . 6 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
5	4	biimpar 281	. . . . 5 |- ( ( F Fn A /\ B C_ A ) -> B C_ dom F )
6		df-ss 2931	. . . . 5 |- ( B C_ dom F <-> ( B i^i dom F ) = B )
7	5, 6	sylib 127	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( B i^i dom F ) = B )
8	2, 7	syl5eq 2084	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( dom F i^i B ) = B )
9	8	eqeq1d 2048	. 2 |- ( ( F Fn A /\ B C_ A ) -> ( ( dom F i^i B ) = (/) <-> B = (/) ) )
10	1, 9	syl5bb 181	1 |- ( ( F Fn A /\ B C_ A ) -> ( ( F " B ) = (/) <-> B = (/) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   i^i cin 2916   C_ wss 2917   (/)c0 3224   dom cdm 4345   "cima 4348   Fn wfn 4897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fn 4905

This theorem is referenced by: (None)