Theorem funimass1 5091

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 4808	. 2 |- ( ( `' F " A ) C_ B -> ( F " ( `' F " A ) ) C_ ( F " B ) )
2		funimacnv 5090	. . . 4 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )
3		dfss 3013	. . . . . 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 2093	. . . 4 |- ( A C_ ran F -> ( A i^i ran F ) = A )
6	2, 5	sylan9eq 2140	. . 3 |- ( ( Fun F /\ A C_ ran F ) -> ( F " ( `' F " A ) ) = A )
7	6	sseq1d 3053	. 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 1289   i^i cin 2998   C_ wss 2999   `'ccnv 4437   ran crn 4439   "cima 4441   Fun wfun 5009

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017

This theorem is referenced by: (None)