Theorem fmucnd 22096

Description: The image of a Cauchy filter base by an uniformly continuous function is a Cauchy filter base. Deduction form. Proposition 3 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 18-Nov-2017.)

Hypotheses

Ref	Expression
fmucnd.1	|- ( ph -> U e. (UnifOn ` X ) )
fmucnd.2	|- ( ph -> V e. (UnifOn ` Y ) )
fmucnd.3	|- ( ph -> F e. ( U Cnu V ) )
fmucnd.4	|- ( ph -> C e. (CauFilu ` U ) )
fmucnd.5	|- D = ran ( a e. C |-> ( F " a ) )

Assertion

Ref	Expression
fmucnd	|- ( ph -> D e. (CauFilu ` V ) )

Distinct variable groups:   C, a   D, a   F, a   V, a   X, a   Y, a   ph, a

Allowed substitution hint:   U( a)

Proof of Theorem fmucnd

Dummy variables c b v r s t x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmucnd.1	. . . 4 |- ( ph -> U e. (UnifOn ` X ) )
2		fmucnd.4	. . . 4 |- ( ph -> C e. (CauFilu ` U ) )
3		cfilufbas 22093	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ C e. (CauFilu ` U ) ) -> C e. ( fBas ` X ) )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ph -> C e. ( fBas ` X ) )
5		fmucnd.2	. . . 4 |- ( ph -> V e. (UnifOn ` Y ) )
6		fmucnd.3	. . . 4 |- ( ph -> F e. ( U Cnu V ) )
7		isucn 22082	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ V e. (UnifOn ` Y ) ) -> ( F e. ( U Cnu V ) <-> ( F : X --> Y /\ A. v e. V E. r e. U A. x e. X A. y e. X ( x r y -> ( F ` x ) v ( F ` y ) ) ) ) )
8	7	simprbda 653	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ V e. (UnifOn ` Y ) ) /\ F e. ( U Cnu V ) ) -> F : X --> Y )
9	1, 5, 6, 8	syl21anc 1325	. . 3 |- ( ph -> F : X --> Y )
10	5	elfvexd 6222	. . 3 |- ( ph -> Y e. _V )
11		fmucnd.5	. . . 4 |- D = ran ( a e. C |-> ( F " a ) )
12	11	fbasrn 21688	. . 3 |- ( ( C e. ( fBas ` X ) /\ F : X --> Y /\ Y e. _V ) -> D e. ( fBas ` Y ) )
13	4, 9, 10, 12	syl3anc 1326	. 2 |- ( ph -> D e. ( fBas ` Y ) )
14		simplr 792	. . . . . . . 8 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> a e. C )
15		eqid 2622	. . . . . . . 8 |- ( F " a ) = ( F " a )
16		imaeq2 5462	. . . . . . . . . 10 |- ( c = a -> ( F " c ) = ( F " a ) )
17	16	eqeq2d 2632	. . . . . . . . 9 |- ( c = a -> ( ( F " a ) = ( F " c ) <-> ( F " a ) = ( F " a ) ) )
18	17	rspcev 3309	. . . . . . . 8 |- ( ( a e. C /\ ( F " a ) = ( F " a ) ) -> E. c e. C ( F " a ) = ( F " c ) )
19	14, 15, 18	sylancl 694	. . . . . . 7 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> E. c e. C ( F " a ) = ( F " c ) )
20		imaexg 7103	. . . . . . . . 9 |- ( F e. ( U Cnu V ) -> ( F " a ) e. _V )
21		eqid 2622	. . . . . . . . . 10 |- ( c e. C |-> ( F " c ) ) = ( c e. C |-> ( F " c ) )
22	21	elrnmpt 5372	. . . . . . . . 9 |- ( ( F " a ) e. _V -> ( ( F " a ) e. ran ( c e. C |-> ( F " c ) ) <-> E. c e. C ( F " a ) = ( F " c ) ) )
23	6, 20, 22	3syl 18	. . . . . . . 8 |- ( ph -> ( ( F " a ) e. ran ( c e. C |-> ( F " c ) ) <-> E. c e. C ( F " a ) = ( F " c ) ) )
24	23	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> ( ( F " a ) e. ran ( c e. C |-> ( F " c ) ) <-> E. c e. C ( F " a ) = ( F " c ) ) )
25	19, 24	mpbird 247	. . . . . 6 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> ( F " a ) e. ran ( c e. C |-> ( F " c ) ) )
26		imaeq2 5462	. . . . . . . . 9 |- ( a = c -> ( F " a ) = ( F " c ) )
27	26	cbvmptv 4750	. . . . . . . 8 |- ( a e. C |-> ( F " a ) ) = ( c e. C |-> ( F " c ) )
28	27	rneqi 5352	. . . . . . 7 |- ran ( a e. C |-> ( F " a ) ) = ran ( c e. C |-> ( F " c ) )
29	11, 28	eqtri 2644	. . . . . 6 |- D = ran ( c e. C |-> ( F " c ) )
30	25, 29	syl6eleqr 2712	. . . . 5 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> ( F " a ) e. D )
31		ffn 6045	. . . . . . . . 9 |- ( F : X --> Y -> F Fn X )
32	9, 31	syl 17	. . . . . . . 8 |- ( ph -> F Fn X )
33	32	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> F Fn X )
34		simplll 798	. . . . . . . 8 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> ph )
35		fbelss 21637	. . . . . . . . 9 |- ( ( C e. ( fBas ` X ) /\ a e. C ) -> a C_ X )
36	4, 35	sylan 488	. . . . . . . 8 |- ( ( ph /\ a e. C ) -> a C_ X )
37	34, 14, 36	syl2anc 693	. . . . . . 7 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> a C_ X )
38		fmucndlem 22095	. . . . . . 7 |- ( ( F Fn X /\ a C_ X ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( a X. a ) ) = ( ( F " a ) X. ( F " a ) ) )
39	33, 37, 38	syl2anc 693	. . . . . 6 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( a X. a ) ) = ( ( F " a ) X. ( F " a ) ) )
40		eqid 2622	. . . . . . . . 9 |- ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) = ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. )
41	40	mpt2fun 6762	. . . . . . . 8 |- Fun ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. )
42		funimass2 5972	. . . . . . . 8 |- ( ( Fun ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( a X. a ) ) C_ v )
43	41, 42	mpan 706	. . . . . . 7 |- ( ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( a X. a ) ) C_ v )
44	43	adantl 482	. . . . . 6 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> ( ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " ( a X. a ) ) C_ v )
45	39, 44	eqsstr3d 3640	. . . . 5 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> ( ( F " a ) X. ( F " a ) ) C_ v )
46		id 22	. . . . . . . 8 |- ( b = ( F " a ) -> b = ( F " a ) )
47	46	sqxpeqd 5141	. . . . . . 7 |- ( b = ( F " a ) -> ( b X. b ) = ( ( F " a ) X. ( F " a ) ) )
48	47	sseq1d 3632	. . . . . 6 |- ( b = ( F " a ) -> ( ( b X. b ) C_ v <-> ( ( F " a ) X. ( F " a ) ) C_ v ) )
49	48	rspcev 3309	. . . . 5 |- ( ( ( F " a ) e. D /\ ( ( F " a ) X. ( F " a ) ) C_ v ) -> E. b e. D ( b X. b ) C_ v )
50	30, 45, 49	syl2anc 693	. . . 4 |- ( ( ( ( ph /\ v e. V ) /\ a e. C ) /\ ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) ) -> E. b e. D ( b X. b ) C_ v )
51	1	adantr 481	. . . . 5 |- ( ( ph /\ v e. V ) -> U e. (UnifOn ` X ) )
52	2	adantr 481	. . . . 5 |- ( ( ph /\ v e. V ) -> C e. (CauFilu ` U ) )
53	5	adantr 481	. . . . . 6 |- ( ( ph /\ v e. V ) -> V e. (UnifOn ` Y ) )
54	6	adantr 481	. . . . . 6 |- ( ( ph /\ v e. V ) -> F e. ( U Cnu V ) )
55		simpr 477	. . . . . 6 |- ( ( ph /\ v e. V ) -> v e. V )
56		nfcv 2764	. . . . . . 7 |- F/_ s <. ( F ` x ) , ( F ` y ) >.
57		nfcv 2764	. . . . . . 7 |- F/_ t <. ( F ` x ) , ( F ` y ) >.
58		nfcv 2764	. . . . . . 7 |- F/_ x <. ( F ` s ) , ( F ` t ) >.
59		nfcv 2764	. . . . . . 7 |- F/_ y <. ( F ` s ) , ( F ` t ) >.
60		simpl 473	. . . . . . . . 9 |- ( ( x = s /\ y = t ) -> x = s )
61	60	fveq2d 6195	. . . . . . . 8 |- ( ( x = s /\ y = t ) -> ( F ` x ) = ( F ` s ) )
62		simpr 477	. . . . . . . . 9 |- ( ( x = s /\ y = t ) -> y = t )
63	62	fveq2d 6195	. . . . . . . 8 |- ( ( x = s /\ y = t ) -> ( F ` y ) = ( F ` t ) )
64	61, 63	opeq12d 4410	. . . . . . 7 |- ( ( x = s /\ y = t ) -> <. ( F ` x ) , ( F ` y ) >. = <. ( F ` s ) , ( F ` t ) >. )
65	56, 57, 58, 59, 64	cbvmpt2 6734	. . . . . 6 |- ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) = ( s e. X , t e. X |-> <. ( F ` s ) , ( F ` t ) >. )
66	51, 53, 54, 55, 65	ucnprima 22086	. . . . 5 |- ( ( ph /\ v e. V ) -> ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) e. U )
67		cfiluexsm 22094	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ C e. (CauFilu ` U ) /\ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) e. U ) -> E. a e. C ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) )
68	51, 52, 66, 67	syl3anc 1326	. . . 4 |- ( ( ph /\ v e. V ) -> E. a e. C ( a X. a ) C_ ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) )
69	50, 68	r19.29a 3078	. . 3 |- ( ( ph /\ v e. V ) -> E. b e. D ( b X. b ) C_ v )
70	69	ralrimiva 2966	. 2 |- ( ph -> A. v e. V E. b e. D ( b X. b ) C_ v )
71		iscfilu 22092	. . 3 |- ( V e. (UnifOn ` Y ) -> ( D e. (CauFilu ` V ) <-> ( D e. ( fBas ` Y ) /\ A. v e. V E. b e. D ( b X. b ) C_ v ) ) )
72	5, 71	syl 17	. 2 |- ( ph -> ( D e. (CauFilu ` V ) <-> ( D e. ( fBas ` Y ) /\ A. v e. V E. b e. D ( b X. b ) C_ v ) ) )
73	13, 70, 72	mpbir2and 957	1 |- ( ph -> D e. (CauFilu ` V ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   fBascfbas 19734  UnifOncust 22003   Cn_ucucn 22079  CauFil_uccfilu 22090

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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-fbas 19743  df-ust 22004  df-ucn 22080  df-cfilu 22091

This theorem is referenced by:  ucnextcn  22108