Theorem funimass1 5438

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 5143	. 2 |- ( ( `' F " A ) C_ B -> ( F " ( `' F " A ) ) C_ ( F " B ) )
2		funimacnv 5437	. . . 4 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )
3		dfss 3228	. . . . . 6 |- ( A C_ ran F <-> A = ( A i^i ran F ) )
4	3	biimpi 120	. . . . 5 |- ( A C_ ran F -> A = ( A i^i ran F ) )
5	4	eqcomd 2240	. . . 4 |- ( A C_ ran F -> ( A i^i ran F ) = A )
6	2, 5	sylan9eq 2287	. . 3 |- ( ( Fun F /\ A C_ ran F ) -> ( F " ( `' F " A ) ) = A )
7	6	sseq1d 3271	. 2 |- ( ( Fun F /\ A C_ ran F ) -> ( ( F " ( `' F " A ) ) C_ ( F " B ) <-> A C_ ( F " B ) ) )
8	1, 7	imbitrid 154	1 |- ( ( Fun F /\ A C_ ran F ) -> ( ( `' F " A ) C_ B -> A C_ ( F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   i^i cin 3213   C_ wss 3214   `'ccnv 4753   ran crn 4755   "cima 4757   Fun wfun 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-fun 5359

This theorem is referenced by:  hmeontr  15304