Theorem funimass2 5405

Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)

Assertion

Ref	Expression
funimass2	|- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B )

Proof of Theorem funimass2

Step	Hyp	Ref	Expression
1		imass2 5110	. 2 |- ( A C_ ( `' F " B ) -> ( F " A ) C_ ( F " ( `' F " B ) ) )
2		funimacnv 5403	. . . . 5 |- ( Fun F -> ( F " ( `' F " B ) ) = ( B i^i ran F ) )
3	2	sseq2d 3255	. . . 4 |- ( Fun F -> ( ( F " A ) C_ ( F " ( `' F " B ) ) <-> ( F " A ) C_ ( B i^i ran F ) ) )
4		inss1 3425	. . . . 5 |- ( B i^i ran F ) C_ B
5		sstr2 3232	. . . . 5 |- ( ( F " A ) C_ ( B i^i ran F ) -> ( ( B i^i ran F ) C_ B -> ( F " A ) C_ B ) )
6	4, 5	mpi 15	. . . 4 |- ( ( F " A ) C_ ( B i^i ran F ) -> ( F " A ) C_ B )
7	3, 6	biimtrdi 163	. . 3 |- ( Fun F -> ( ( F " A ) C_ ( F " ( `' F " B ) ) -> ( F " A ) C_ B ) )
8	7	imp 124	. 2 |- ( ( Fun F /\ ( F " A ) C_ ( F " ( `' F " B ) ) ) -> ( F " A ) C_ B )
9	1, 8	sylan2 286	1 |- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B )

Syntax hints:   -> wi 4   /\ wa 104   i^i cin 3197   C_ wss 3198   `'ccnv 4722   ran crn 4724   "cima 4726   Fun wfun 5318

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fun 5326

This theorem is referenced by:  fvimacnvi  5757