Theorem funimass2 4997

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 4721	. 2 |- ( A C_ ( `' F " B ) -> ( F " A ) C_ ( F " ( `' F " B ) ) )
2		funimacnv 4995	. . . . 5 |- ( Fun F -> ( F " ( `' F " B ) ) = ( B i^i ran F ) )
3	2	sseq2d 3027	. . . 4 |- ( Fun F -> ( ( F " A ) C_ ( F " ( `' F " B ) ) <-> ( F " A ) C_ ( B i^i ran F ) ) )
4		inss1 3186	. . . . 5 |- ( B i^i ran F ) C_ B
5		sstr2 3006	. . . . 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	syl6bi 161	. . 3 |- ( Fun F -> ( ( F " A ) C_ ( F " ( `' F " B ) ) -> ( F " A ) C_ B ) )
8	7	imp 122	. 2 |- ( ( Fun F /\ ( F " A ) C_ ( F " ( `' F " B ) ) ) -> ( F " A ) C_ B )
9	1, 8	sylan2 280	1 |- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B )

Syntax hints:   -> wi 4   /\ wa 102   i^i cin 2972   C_ wss 2973   `'ccnv 4362   ran crn 4364   "cima 4366   Fun wfun 4916

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-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-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-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-fun 4924

This theorem is referenced by:  fvimacnvi  5302