Theorem fmucnd 19842

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 19839	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ C e. (CauFilu ` U ) ) -> C e. ( fBas ` X ) )
4	1, 2, 3	syl2anc 661	. . 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 19828	. . . . 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 623	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ V e. (UnifOn ` Y ) ) /\ F e. ( U Cnu V ) ) -> F : X --> Y )
9	1, 5, 6, 8	syl21anc 1217	. . 3 |- ( ph -> F : X --> Y )
10	5	elfvexd 5713	. . 3 |- ( ph -> Y e. _V )
11		fmucnd.5	. . . 4 |- D = ran ( a e. C |-> ( F " a ) )
12	11	fbasrn 19432	. . 3 |- ( ( C e. ( fBas ` X ) /\ F : X --> Y /\ Y e. _V ) -> D e. ( fBas ` Y ) )
13	4, 9, 10, 12	syl3anc 1218	. 2 |- ( ph -> D e. ( fBas ` Y ) )
14		simplr 754	. . . . . . . 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 2438	. . . . . . . 8 |- ( F " a ) = ( F " a )
16		imaeq2 5160	. . . . . . . . . 10 |- ( c = a -> ( F " c ) = ( F " a ) )
17	16	eqeq2d 2449	. . . . . . . . 9 |- ( c = a -> ( ( F " a ) = ( F " c ) <-> ( F " a ) = ( F " a ) ) )
18	17	rspcev 3068	. . . . . . . 8 |- ( ( a e. C /\ ( F " a ) = ( F " a ) ) -> E. c e. C ( F " a ) = ( F " c ) )
19	14, 15, 18	sylancl 662	. . . . . . 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 6510	. . . . . . . . 9 |- ( F e. ( U Cnu V ) -> ( F " a ) e. _V )
21		eqid 2438	. . . . . . . . . 10 |- ( c e. C |-> ( F " c ) ) = ( c e. C |-> ( F " c ) )
22	21	elrnmpt 5081	. . . . . . . . 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 20	. . . . . . . 8 |- ( ph -> ( ( F " a ) e. ran ( c e. C |-> ( F " c ) ) <-> E. c e. C ( F " a ) = ( F " c ) ) )
24	23	ad3antrrr 729	. . . . . . 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 232	. . . . . 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 5160	. . . . . . . . 9 |- ( a = c -> ( F " a ) = ( F " c ) )
27	26	cbvmptv 4378	. . . . . . . 8 |- ( a e. C |-> ( F " a ) ) = ( c e. C |-> ( F " c ) )
28	27	rneqi 5061	. . . . . . 7 |- ran ( a e. C |-> ( F " a ) ) = ran ( c e. C |-> ( F " c ) )
29	11, 28	eqtri 2458	. . . . . 6 |- D = ran ( c e. C |-> ( F " c ) )
30	25, 29	syl6eleqr 2529	. . . . 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 5554	. . . . . . . . 9 |- ( F : X --> Y -> F Fn X )
32	9, 31	syl 16	. . . . . . . 8 |- ( ph -> F Fn X )
33	32	ad3antrrr 729	. . . . . . 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 757	. . . . . . . 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 19381	. . . . . . . . 9 |- ( ( C e. ( fBas ` X ) /\ a e. C ) -> a C_ X )
36	4, 35	sylan 471	. . . . . . . 8 |- ( ( ph /\ a e. C ) -> a C_ X )
37	34, 14, 36	syl2anc 661	. . . . . . 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 19841	. . . . . . 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 661	. . . . . 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 2438	. . . . . . . . 9 |- ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) = ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. )
41	40	mpt2fun 6187	. . . . . . . 8 |- Fun ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. )
42		funimass2 5487	. . . . . . . 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 670	. . . . . . 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 466	. . . . . 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 3386	. . . . 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, 46	xpeq12d 4860	. . . . . . 7 |- ( b = ( F " a ) -> ( b X. b ) = ( ( F " a ) X. ( F " a ) ) )
48	47	sseq1d 3378	. . . . . 6 |- ( b = ( F " a ) -> ( ( b X. b ) C_ v <-> ( ( F " a ) X. ( F " a ) ) C_ v ) )
49	48	rspcev 3068	. . . . 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 661	. . . 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 465	. . . . 5 |- ( ( ph /\ v e. V ) -> U e. (UnifOn ` X ) )
52	2	adantr 465	. . . . 5 |- ( ( ph /\ v e. V ) -> C e. (CauFilu ` U ) )
53	5	adantr 465	. . . . . 6 |- ( ( ph /\ v e. V ) -> V e. (UnifOn ` Y ) )
54	6	adantr 465	. . . . . 6 |- ( ( ph /\ v e. V ) -> F e. ( U Cnu V ) )
55		simpr 461	. . . . . 6 |- ( ( ph /\ v e. V ) -> v e. V )
56		nfcv 2574	. . . . . . 7 |- F/_ s <. ( F ` x ) , ( F ` y ) >.
57		nfcv 2574	. . . . . . 7 |- F/_ t <. ( F ` x ) , ( F ` y ) >.
58		nfcv 2574	. . . . . . 7 |- F/_ x <. ( F ` s ) , ( F ` t ) >.
59		nfcv 2574	. . . . . . 7 |- F/_ y <. ( F ` s ) , ( F ` t ) >.
60		simpl 457	. . . . . . . . 9 |- ( ( x = s /\ y = t ) -> x = s )
61	60	fveq2d 5690	. . . . . . . 8 |- ( ( x = s /\ y = t ) -> ( F ` x ) = ( F ` s ) )
62		simpr 461	. . . . . . . . 9 |- ( ( x = s /\ y = t ) -> y = t )
63	62	fveq2d 5690	. . . . . . . 8 |- ( ( x = s /\ y = t ) -> ( F ` y ) = ( F ` t ) )
64	61, 63	opeq12d 4062	. . . . . . 7 |- ( ( x = s /\ y = t ) -> <. ( F ` x ) , ( F ` y ) >. = <. ( F ` s ) , ( F ` t ) >. )
65	56, 57, 58, 59, 64	cbvmpt2 6160	. . . . . 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 19832	. . . . 5 |- ( ( ph /\ v e. V ) -> ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) e. U )
67		cfiluexsm 19840	. . . . 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 1218	. . . 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 2857	. . 3 |- ( ( ph /\ v e. V ) -> E. b e. D ( b X. b ) C_ v )
70	69	ralrimiva 2794	. 2 |- ( ph -> A. v e. V E. b e. D ( b X. b ) C_ v )
71		iscfilu 19838	. . 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 16	. 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 913	1 |- ( ph -> D e. (CauFilu ` V ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2710   E.wrex 2711   _Vcvv 2967   C_ wss 3323   <.cop 3878   class class class wbr 4287   e. cmpt 4345   X. cxp 4833   `'ccnv 4834   ran crn 4836   "cima 4838   Fun wfun 5407   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   e. cmpt2 6088   fBascfbas 17779  UnifOncust 19749   Cn_ucucn 19825  CauFil_uccfilu 19836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-map 7208  df-fbas 17789  df-ust 19750  df-ucn 19826  df-cfilu 19837

This theorem is referenced by:  ucnextcn  19854