Theorem elfm 21751

Description: An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

Ref	Expression
elfm	|- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( A e. ( ( X FilMap F ) ` B ) <-> ( A C_ X /\ E. x e. B ( F " x ) C_ A ) ) )

Distinct variable groups:   x, B   x, C   x, F   x, X   x, A   x, Y

Proof of Theorem elfm

Dummy variables t y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmval 21747	. . 3 |- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` B ) = ( X filGen ran ( t e. B |-> ( F " t ) ) ) )
2	1	eleq2d 2687	. 2 |- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( A e. ( ( X FilMap F ) ` B ) <-> A e. ( X filGen ran ( t e. B |-> ( F " t ) ) ) ) )
3		eqid 2622	. . . . 5 |- ran ( t e. B |-> ( F " t ) ) = ran ( t e. B |-> ( F " t ) )
4	3	fbasrn 21688	. . . 4 |- ( ( B e. ( fBas ` Y ) /\ F : Y --> X /\ X e. C ) -> ran ( t e. B |-> ( F " t ) ) e. ( fBas ` X ) )
5	4	3comr 1273	. . 3 |- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ran ( t e. B |-> ( F " t ) ) e. ( fBas ` X ) )
6		elfg 21675	. . 3 |- ( ran ( t e. B |-> ( F " t ) ) e. ( fBas ` X ) -> ( A e. ( X filGen ran ( t e. B |-> ( F " t ) ) ) <-> ( A C_ X /\ E. y e. ran ( t e. B |-> ( F " t ) ) y C_ A ) ) )
7	5, 6	syl 17	. 2 |- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( A e. ( X filGen ran ( t e. B |-> ( F " t ) ) ) <-> ( A C_ X /\ E. y e. ran ( t e. B |-> ( F " t ) ) y C_ A ) ) )
8		simpr 477	. . . . . 6 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ x e. B ) -> x e. B )
9		eqid 2622	. . . . . 6 |- ( F " x ) = ( F " x )
10		imaeq2 5462	. . . . . . . 8 |- ( t = x -> ( F " t ) = ( F " x ) )
11	10	eqeq2d 2632	. . . . . . 7 |- ( t = x -> ( ( F " x ) = ( F " t ) <-> ( F " x ) = ( F " x ) ) )
12	11	rspcev 3309	. . . . . 6 |- ( ( x e. B /\ ( F " x ) = ( F " x ) ) -> E. t e. B ( F " x ) = ( F " t ) )
13	8, 9, 12	sylancl 694	. . . . 5 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ x e. B ) -> E. t e. B ( F " x ) = ( F " t ) )
14		simpl1 1064	. . . . . . 7 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ x e. B ) -> X e. C )
15		imassrn 5477	. . . . . . . 8 |- ( F " x ) C_ ran F
16		frn 6053	. . . . . . . . . 10 |- ( F : Y --> X -> ran F C_ X )
17	16	3ad2ant3 1084	. . . . . . . . 9 |- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ran F C_ X )
18	17	adantr 481	. . . . . . . 8 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ x e. B ) -> ran F C_ X )
19	15, 18	syl5ss 3614	. . . . . . 7 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ x e. B ) -> ( F " x ) C_ X )
20	14, 19	ssexd 4805	. . . . . 6 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ x e. B ) -> ( F " x ) e. _V )
21		eqid 2622	. . . . . . 7 |- ( t e. B |-> ( F " t ) ) = ( t e. B |-> ( F " t ) )
22	21	elrnmpt 5372	. . . . . 6 |- ( ( F " x ) e. _V -> ( ( F " x ) e. ran ( t e. B |-> ( F " t ) ) <-> E. t e. B ( F " x ) = ( F " t ) ) )
23	20, 22	syl 17	. . . . 5 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ x e. B ) -> ( ( F " x ) e. ran ( t e. B |-> ( F " t ) ) <-> E. t e. B ( F " x ) = ( F " t ) ) )
24	13, 23	mpbird 247	. . . 4 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ x e. B ) -> ( F " x ) e. ran ( t e. B |-> ( F " t ) ) )
25	10	cbvmptv 4750	. . . . . . 7 |- ( t e. B |-> ( F " t ) ) = ( x e. B |-> ( F " x ) )
26	25	elrnmpt 5372	. . . . . 6 |- ( y e. ran ( t e. B |-> ( F " t ) ) -> ( y e. ran ( t e. B |-> ( F " t ) ) <-> E. x e. B y = ( F " x ) ) )
27	26	ibi 256	. . . . 5 |- ( y e. ran ( t e. B |-> ( F " t ) ) -> E. x e. B y = ( F " x ) )
28	27	adantl 482	. . . 4 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ y e. ran ( t e. B |-> ( F " t ) ) ) -> E. x e. B y = ( F " x ) )
29		simpr 477	. . . . 5 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ y = ( F " x ) ) -> y = ( F " x ) )
30	29	sseq1d 3632	. . . 4 |- ( ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) /\ y = ( F " x ) ) -> ( y C_ A <-> ( F " x ) C_ A ) )
31	24, 28, 30	rexxfrd 4881	. . 3 |- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( E. y e. ran ( t e. B |-> ( F " t ) ) y C_ A <-> E. x e. B ( F " x ) C_ A ) )
32	31	anbi2d 740	. 2 |- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( A C_ X /\ E. y e. ran ( t e. B |-> ( F " t ) ) y C_ A ) <-> ( A C_ X /\ E. x e. B ( F " x ) C_ A ) ) )
33	2, 7, 32	3bitrd 294	1 |- ( ( X e. C /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( A e. ( ( X FilMap F ) ` B ) <-> ( A C_ X /\ E. x e. B ( F " x ) C_ A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   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:  elfm2  21752  fmfg  21753  rnelfm  21757  fmfnfmlem1  21758  fmfnfm  21762  fmco  21765  flfnei  21795  isflf  21797  isfcf  21838  filnetlem4  32376