Theorem funimass1 4996

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

Assertion

Ref	Expression
funimass1	|- ( ( Fun F /\ A C_ ran F ) -> ( ( `' F " A ) C_ B -> A C_ ( F " B ) ) )

Proof of Theorem funimass1

Step	Hyp	Ref	Expression
1		imass2 4721	. 2 |- ( ( `' F " A ) C_ B -> ( F " ( `' F " A ) ) C_ ( F " B ) )
2		funimacnv 4995	. . . 4 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )
3		dfss 2987	. . . . . 6 |- ( A C_ ran F <-> A = ( A i^i ran F ) )
4	3	biimpi 118	. . . . 5 |- ( A C_ ran F -> A = ( A i^i ran F ) )
5	4	eqcomd 2086	. . . 4 |- ( A C_ ran F -> ( A i^i ran F ) = A )
6	2, 5	sylan9eq 2133	. . 3 |- ( ( Fun F /\ A C_ ran F ) -> ( F " ( `' F " A ) ) = A )
7	6	sseq1d 3026	. 2 |- ( ( Fun F /\ A C_ ran F ) -> ( ( F " ( `' F " A ) ) C_ ( F " B ) <-> A C_ ( F " B ) ) )
8	1, 7	syl5ib 152	1 |- ( ( Fun F /\ A C_ ran F ) -> ( ( `' F " A ) C_ B -> A C_ ( F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   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: (None)