Theorem funimass2 5454

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 5158	. 2 |- ( A C_ ( `' F " B ) -> ( F " A ) C_ ( F " ( `' F " B ) ) )
2		funimacnv 5452	. . . . 5 |- ( Fun F -> ( F " ( `' F " B ) ) = ( B i^i ran F ) )
3	2	sseq2d 3278	. . . 4 |- ( Fun F -> ( ( F " A ) C_ ( F " ( `' F " B ) ) <-> ( F " A ) C_ ( B i^i ran F ) ) )
4		inss1 3451	. . . . 5 |- ( B i^i ran F ) C_ B
5		sstr2 3255	. . . . 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 3219   C_ wss 3220   `'ccnv 4768   ran crn 4770   "cima 4772   Fun wfun 5366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374

This theorem is referenced by:  fvimacnvi  5814