Theorem imaelfm 21755

Description: An image of a filter element is in the image filter. (Contributed by Jeff Hankins, 5-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
imaelfm.l	|- L = ( Y filGen B )

Assertion

Ref	Expression
imaelfm	|- ( ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ S e. L ) -> ( F " S ) e. ( ( X FilMap F ) ` B ) )

Proof of Theorem imaelfm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 5477	. . . . 5 |- ( F " S ) C_ ran F
2		frn 6053	. . . . 5 |- ( F : Y --> X -> ran F C_ X )
3	1, 2	syl5ss 3614	. . . 4 |- ( F : Y --> X -> ( F " S ) C_ X )
4	3	3ad2ant3 1084	. . 3 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( F " S ) C_ X )
5		ssid 3624	. . . 4 |- ( F " S ) C_ ( F " S )
6		imaeq2 5462	. . . . . 6 |- ( x = S -> ( F " x ) = ( F " S ) )
7	6	sseq1d 3632	. . . . 5 |- ( x = S -> ( ( F " x ) C_ ( F " S ) <-> ( F " S ) C_ ( F " S ) ) )
8	7	rspcev 3309	. . . 4 |- ( ( S e. L /\ ( F " S ) C_ ( F " S ) ) -> E. x e. L ( F " x ) C_ ( F " S ) )
9	5, 8	mpan2 707	. . 3 |- ( S e. L -> E. x e. L ( F " x ) C_ ( F " S ) )
10	4, 9	anim12i 590	. 2 |- ( ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ S e. L ) -> ( ( F " S ) C_ X /\ E. x e. L ( F " x ) C_ ( F " S ) ) )
11		imaelfm.l	. . . 4 |- L = ( Y filGen B )
12	11	elfm2 21752	. . 3 |- ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( F " S ) e. ( ( X FilMap F ) ` B ) <-> ( ( F " S ) C_ X /\ E. x e. L ( F " x ) C_ ( F " S ) ) ) )
13	12	adantr 481	. 2 |- ( ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ S e. L ) -> ( ( F " S ) e. ( ( X FilMap F ) ` B ) <-> ( ( F " S ) C_ X /\ E. x e. L ( F " x ) C_ ( F " S ) ) ) )
14	10, 13	mpbird 247	1 |- ( ( ( X e. A /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ S e. L ) -> ( F " S ) e. ( ( X FilMap F ) ` B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   fBascfbas 19734   filGencfg 19735   FilMap cfm 21737

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fg 19744  df-fm 21742

This theorem is referenced by:  rnelfm  21757  fmfnfmlem2  21759  fmfnfmlem4  21761  fmfnfm  21762  fmco  21765  isfcf  21838  cnextcn  21871