Theorem funimass1 5208

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 4923	. 2 |- ( ( `' F " A ) C_ B -> ( F " ( `' F " A ) ) C_ ( F " B ) )
2		funimacnv 5207	. . . 4 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )
3		dfss 3090	. . . . . 6 |- ( A C_ ran F <-> A = ( A i^i ran F ) )
4	3	biimpi 119	. . . . 5 |- ( A C_ ran F -> A = ( A i^i ran F ) )
5	4	eqcomd 2146	. . . 4 |- ( A C_ ran F -> ( A i^i ran F ) = A )
6	2, 5	sylan9eq 2193	. . 3 |- ( ( Fun F /\ A C_ ran F ) -> ( F " ( `' F " A ) ) = A )
7	6	sseq1d 3131	. 2 |- ( ( Fun F /\ A C_ ran F ) -> ( ( F " ( `' F " A ) ) C_ ( F " B ) <-> A C_ ( F " B ) ) )
8	1, 7	syl5ib 153	1 |- ( ( Fun F /\ A C_ ran F ) -> ( ( `' F " A ) C_ B -> A C_ ( F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   i^i cin 3075   C_ wss 3076   `'ccnv 4546   ran crn 4548   "cima 4550   Fun wfun 5125

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-fun 5133

This theorem is referenced by:  hmeontr  12521