Theorem funimass1 5001

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 4725	. 2 |- ( ( `' F " A ) C_ B -> ( F " ( `' F " A ) ) C_ ( F " B ) )
2		funimacnv 5000	. . . 4 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )
3		dfss 2988	. . . . . 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 2087	. . . 4 |- ( A C_ ran F -> ( A i^i ran F ) = A )
6	2, 5	sylan9eq 2134	. . 3 |- ( ( Fun F /\ A C_ ran F ) -> ( F " ( `' F " A ) ) = A )
7	6	sseq1d 3027	. 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 1285   i^i cin 2973   C_ wss 2974   `'ccnv 4364   ran crn 4366   "cima 4368   Fun wfun 4920

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-fun 4928

This theorem is referenced by: (None)