Theorem funimass2 5972

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 5501	. 2 |- ( A C_ ( `' F " B ) -> ( F " A ) C_ ( F " ( `' F " B ) ) )
2		funimacnv 5970	. . . . 5 |- ( Fun F -> ( F " ( `' F " B ) ) = ( B i^i ran F ) )
3	2	sseq2d 3633	. . . 4 |- ( Fun F -> ( ( F " A ) C_ ( F " ( `' F " B ) ) <-> ( F " A ) C_ ( B i^i ran F ) ) )
4		inss1 3833	. . . . 5 |- ( B i^i ran F ) C_ B
5		sstr2 3610	. . . . 5 |- ( ( F " A ) C_ ( B i^i ran F ) -> ( ( B i^i ran F ) C_ B -> ( F " A ) C_ B ) )
6	4, 5	mpi 20	. . . 4 |- ( ( F " A ) C_ ( B i^i ran F ) -> ( F " A ) C_ B )
7	3, 6	syl6bi 243	. . 3 |- ( Fun F -> ( ( F " A ) C_ ( F " ( `' F " B ) ) -> ( F " A ) C_ B ) )
8	7	imp 445	. 2 |- ( ( Fun F /\ ( F " A ) C_ ( F " ( `' F " B ) ) ) -> ( F " A ) C_ B )
9	1, 8	sylan2 491	1 |- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B )

Syntax hints:   -> wi 4   /\ wa 384   i^i cin 3573   C_ wss 3574   `'ccnv 5113   ran crn 5115   "cima 5117   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890

This theorem is referenced by:  fvimacnvi  6331  lmhmlsp  19049  2ndcomap  21261  tgqtop  21515  kqreglem1  21544  fmfnfmlem4  21761  fmucnd  22096  cfilucfil  22364