Theorem fmfnfmlem1 21758

Description: Lemma for fmfnfm 21762. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypotheses

Ref	Expression
fmfnfm.b	|- ( ph -> B e. ( fBas ` Y ) )
fmfnfm.l	|- ( ph -> L e. ( Fil ` X ) )
fmfnfm.f	|- ( ph -> F : Y --> X )
fmfnfm.fm	|- ( ph -> ( ( X FilMap F ) ` B ) C_ L )

Assertion

Ref	Expression
fmfnfmlem1	|- ( ph -> ( s e. ( fi ` B ) -> ( ( F " s ) C_ t -> ( t C_ X -> t e. L ) ) ) )

Distinct variable groups:   t, s, B   F, s, t   L, s, t   ph, s, t   X, s, t   Y, s, t

Proof of Theorem fmfnfmlem1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmfnfm.b	. . . . 5 |- ( ph -> B e. ( fBas ` Y ) )
2		fbssfi 21641	. . . . 5 |- ( ( B e. ( fBas ` Y ) /\ s e. ( fi ` B ) ) -> E. w e. B w C_ s )
3	1, 2	sylan 488	. . . 4 |- ( ( ph /\ s e. ( fi ` B ) ) -> E. w e. B w C_ s )
4		sstr2 3610	. . . . . 6 |- ( ( F " w ) C_ ( F " s ) -> ( ( F " s ) C_ t -> ( F " w ) C_ t ) )
5		imass2 5501	. . . . . 6 |- ( w C_ s -> ( F " w ) C_ ( F " s ) )
6	4, 5	syl11 33	. . . . 5 |- ( ( F " s ) C_ t -> ( w C_ s -> ( F " w ) C_ t ) )
7	6	reximdv 3016	. . . 4 |- ( ( F " s ) C_ t -> ( E. w e. B w C_ s -> E. w e. B ( F " w ) C_ t ) )
8	3, 7	syl5com 31	. . 3 |- ( ( ph /\ s e. ( fi ` B ) ) -> ( ( F " s ) C_ t -> E. w e. B ( F " w ) C_ t ) )
9		fmfnfm.l	. . . . . . . 8 |- ( ph -> L e. ( Fil ` X ) )
10		filtop 21659	. . . . . . . 8 |- ( L e. ( Fil ` X ) -> X e. L )
11	9, 10	syl 17	. . . . . . 7 |- ( ph -> X e. L )
12		fmfnfm.f	. . . . . . 7 |- ( ph -> F : Y --> X )
13		elfm 21751	. . . . . . 7 |- ( ( X e. L /\ B e. ( fBas ` Y ) /\ F : Y --> X ) -> ( t e. ( ( X FilMap F ) ` B ) <-> ( t C_ X /\ E. w e. B ( F " w ) C_ t ) ) )
14	11, 1, 12, 13	syl3anc 1326	. . . . . 6 |- ( ph -> ( t e. ( ( X FilMap F ) ` B ) <-> ( t C_ X /\ E. w e. B ( F " w ) C_ t ) ) )
15		fmfnfm.fm	. . . . . . 7 |- ( ph -> ( ( X FilMap F ) ` B ) C_ L )
16	15	sseld 3602	. . . . . 6 |- ( ph -> ( t e. ( ( X FilMap F ) ` B ) -> t e. L ) )
17	14, 16	sylbird 250	. . . . 5 |- ( ph -> ( ( t C_ X /\ E. w e. B ( F " w ) C_ t ) -> t e. L ) )
18	17	expcomd 454	. . . 4 |- ( ph -> ( E. w e. B ( F " w ) C_ t -> ( t C_ X -> t e. L ) ) )
19	18	adantr 481	. . 3 |- ( ( ph /\ s e. ( fi ` B ) ) -> ( E. w e. B ( F " w ) C_ t -> ( t C_ X -> t e. L ) ) )
20	8, 19	syld 47	. 2 |- ( ( ph /\ s e. ( fi ` B ) ) -> ( ( F " s ) C_ t -> ( t C_ X -> t e. L ) ) )
21	20	ex 450	1 |- ( ph -> ( s e. ( fi ` B ) -> ( ( F " s ) C_ t -> ( t C_ X -> t e. L ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   E.wrex 2913   C_ wss 3574   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   ficfi 8316   fBascfbas 19734   Filcfil 21649   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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-fbas 19743  df-fg 19744  df-fil 21650  df-fm 21742

This theorem is referenced by:  fmfnfmlem4  21761