Theorem funimass2 5201

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 4915	. 2 |- ( A C_ ( `' F " B ) -> ( F " A ) C_ ( F " ( `' F " B ) ) )
2		funimacnv 5199	. . . . 5 |- ( Fun F -> ( F " ( `' F " B ) ) = ( B i^i ran F ) )
3	2	sseq2d 3127	. . . 4 |- ( Fun F -> ( ( F " A ) C_ ( F " ( `' F " B ) ) <-> ( F " A ) C_ ( B i^i ran F ) ) )
4		inss1 3296	. . . . 5 |- ( B i^i ran F ) C_ B
5		sstr2 3104	. . . . 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 162	. . 3 |- ( Fun F -> ( ( F " A ) C_ ( F " ( `' F " B ) ) -> ( F " A ) C_ B ) )
8	7	imp 123	. 2 |- ( ( Fun F /\ ( F " A ) C_ ( F " ( `' F " B ) ) ) -> ( F " A ) C_ B )
9	1, 8	sylan2 284	1 |- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B )

Syntax hints:   -> wi 4   /\ wa 103   i^i cin 3070   C_ wss 3071   `'ccnv 4538   ran crn 4540   "cima 4542   Fun wfun 5117

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125

This theorem is referenced by:  fvimacnvi  5534