Theorem fmucnd 18275

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 18272	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ C e. (CauFilu ` U ) ) -> C e. ( fBas ` X ) )
4	1, 2, 3	syl2anc 643	. . 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 18261	. . . . 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 607	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ V e. (UnifOn ` Y ) ) /\ F e. ( U Cnu V ) ) -> F : X --> Y )
9	1, 5, 6, 8	syl21anc 1183	. . 3 |- ( ph -> F : X --> Y )
10	5	elfvexd 5718	. . 3 |- ( ph -> Y e. _V )
11		fmucnd.5	. . . 4 |- D = ran ( a e. C |-> ( F " a ) )
12	11	fbasrn 17869	. . 3 |- ( ( C e. ( fBas ` X ) /\ F : X --> Y /\ Y e. _V ) -> D e. ( fBas ` Y ) )
13	4, 9, 10, 12	syl3anc 1184	. 2 |- ( ph -> D e. ( fBas ` Y ) )
14		simplr 732	. . . . . . . 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 2404	. . . . . . . 8 |- ( F " a ) = ( F " a )
16		imaeq2 5158	. . . . . . . . . 10 |- ( c = a -> ( F " c ) = ( F " a ) )
17	16	eqeq2d 2415	. . . . . . . . 9 |- ( c = a -> ( ( F " a ) = ( F " c ) <-> ( F " a ) = ( F " a ) ) )
18	17	rspcev 3012	. . . . . . . 8 |- ( ( a e. C /\ ( F " a ) = ( F " a ) ) -> E. c e. C ( F " a ) = ( F " c ) )
19	14, 15, 18	sylancl 644	. . . . . . 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 5176	. . . . . . . . 9 |- ( F e. ( U Cnu V ) -> ( F " a ) e. _V )
21		eqid 2404	. . . . . . . . . 10 |- ( c e. C |-> ( F " c ) ) = ( c e. C |-> ( F " c ) )
22	21	elrnmpt 5076	. . . . . . . . 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 19	. . . . . . . 8 |- ( ph -> ( ( F " a ) e. ran ( c e. C |-> ( F " c ) ) <-> E. c e. C ( F " a ) = ( F " c ) ) )
24	23	ad3antrrr 711	. . . . . . 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 224	. . . . . 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 5158	. . . . . . . . 9 |- ( a = c -> ( F " a ) = ( F " c ) )
27	26	cbvmptv 4260	. . . . . . . 8 |- ( a e. C |-> ( F " a ) ) = ( c e. C |-> ( F " c ) )
28	27	rneqi 5055	. . . . . . 7 |- ran ( a e. C |-> ( F " a ) ) = ran ( c e. C |-> ( F " c ) )
29	11, 28	eqtri 2424	. . . . . 6 |- D = ran ( c e. C |-> ( F " c ) )
30	25, 29	syl6eleqr 2495	. . . . 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 5550	. . . . . . . . 9 |- ( F : X --> Y -> F Fn X )
32	9, 31	syl 16	. . . . . . . 8 |- ( ph -> F Fn X )
33	32	ad3antrrr 711	. . . . . . 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 735	. . . . . . . 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 17818	. . . . . . . . 9 |- ( ( C e. ( fBas ` X ) /\ a e. C ) -> a C_ X )
36	4, 35	sylan 458	. . . . . . . 8 |- ( ( ph /\ a e. C ) -> a C_ X )
37	34, 14, 36	syl2anc 643	. . . . . . 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 18274	. . . . . . 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 643	. . . . . 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 2404	. . . . . . . . 9 |- ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) = ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. )
41	40	mpt2fun 6131	. . . . . . . 8 |- Fun ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. )
42		funimass2 5486	. . . . . . . 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 652	. . . . . . 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 453	. . . . . 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 3343	. . . . 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 20	. . . . . . . 8 |- ( b = ( F " a ) -> b = ( F " a ) )
47	46, 46	xpeq12d 4862	. . . . . . 7 |- ( b = ( F " a ) -> ( b X. b ) = ( ( F " a ) X. ( F " a ) ) )
48	47	sseq1d 3335	. . . . . 6 |- ( b = ( F " a ) -> ( ( b X. b ) C_ v <-> ( ( F " a ) X. ( F " a ) ) C_ v ) )
49	48	rspcev 3012	. . . . 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 643	. . . 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 452	. . . . 5 |- ( ( ph /\ v e. V ) -> U e. (UnifOn ` X ) )
52	2	adantr 452	. . . . 5 |- ( ( ph /\ v e. V ) -> C e. (CauFilu ` U ) )
53	5	adantr 452	. . . . . 6 |- ( ( ph /\ v e. V ) -> V e. (UnifOn ` Y ) )
54	6	adantr 452	. . . . . 6 |- ( ( ph /\ v e. V ) -> F e. ( U Cnu V ) )
55		simpr 448	. . . . . 6 |- ( ( ph /\ v e. V ) -> v e. V )
56		nfcv 2540	. . . . . . 7 |- F/_ s <. ( F ` x ) , ( F ` y ) >.
57		nfcv 2540	. . . . . . 7 |- F/_ t <. ( F ` x ) , ( F ` y ) >.
58		nfcv 2540	. . . . . . 7 |- F/_ x <. ( F ` s ) , ( F ` t ) >.
59		nfcv 2540	. . . . . . 7 |- F/_ y <. ( F ` s ) , ( F ` t ) >.
60		simpl 444	. . . . . . . . 9 |- ( ( x = s /\ y = t ) -> x = s )
61	60	fveq2d 5691	. . . . . . . 8 |- ( ( x = s /\ y = t ) -> ( F ` x ) = ( F ` s ) )
62		simpr 448	. . . . . . . . 9 |- ( ( x = s /\ y = t ) -> y = t )
63	62	fveq2d 5691	. . . . . . . 8 |- ( ( x = s /\ y = t ) -> ( F ` y ) = ( F ` t ) )
64	61, 63	opeq12d 3952	. . . . . . 7 |- ( ( x = s /\ y = t ) -> <. ( F ` x ) , ( F ` y ) >. = <. ( F ` s ) , ( F ` t ) >. )
65	56, 57, 58, 59, 64	cbvmpt2 6110	. . . . . 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 18265	. . . . 5 |- ( ( ph /\ v e. V ) -> ( `' ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. ) " v ) e. U )
67		cfiluexsm 18273	. . . . 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 1184	. . . 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 2810	. . 3 |- ( ( ph /\ v e. V ) -> E. b e. D ( b X. b ) C_ v )
70	69	ralrimiva 2749	. 2 |- ( ph -> A. v e. V E. b e. D ( b X. b ) C_ v )
71		iscfilu 18271	. . 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 889	1 |- ( ph -> D e. (CauFilu ` V ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916   C_ wss 3280   <.cop 3777   class class class wbr 4172   e. cmpt 4226   X. cxp 4835   `'ccnv 4836   ran crn 4838   "cima 4840   Fun wfun 5407   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   e. cmpt2 6042   fBascfbas 16644  UnifOncust 18182   Cn_ucucn 18258  CauFil_uccfilu 18269

This theorem is referenced by:  ucnextcn  18287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-map 6979  df-fbas 16654  df-ust 18183  df-ucn 18259  df-cfilu 18270