Theorem cvmlift2lem13 31297

Description: Lemma for cvmlift2 31298. (Contributed by Mario Carneiro, 7-May-2015.)

Hypotheses

Ref	Expression
cvmlift2.b	|- B = U. C
cvmlift2.f	|- ( ph -> F e. ( C CovMap J ) )
cvmlift2.g	|- ( ph -> G e. ( ( II tX II ) Cn J ) )
cvmlift2.p	|- ( ph -> P e. B )
cvmlift2.i	|- ( ph -> ( F ` P ) = ( 0 G 0 ) )
cvmlift2.h	|- H = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( f ` 0 ) = P ) )
cvmlift2.k	|- K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) )

Assertion

Ref	Expression
cvmlift2lem13	|- ( ph -> E! g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) )

Distinct variable groups:   f, g, x, y, z, F   ph, f, g, x, y, z   f, J, g, x, y, z   f, G, g, x, y, z   f, H, x, y, z   C, f, g, x, y, z   P, f, g, x, y, z   x, B, y, z   f, K, g, x, y, z

Allowed substitution hints:   B( f, g)   H( g)

Proof of Theorem cvmlift2lem13

Dummy variables b c d u v a r t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift2.b	. . . 4 |- B = U. C
2		cvmlift2.f	. . . 4 |- ( ph -> F e. ( C CovMap J ) )
3		cvmlift2.g	. . . 4 |- ( ph -> G e. ( ( II tX II ) Cn J ) )
4		cvmlift2.p	. . . 4 |- ( ph -> P e. B )
5		cvmlift2.i	. . . 4 |- ( ph -> ( F ` P ) = ( 0 G 0 ) )
6		cvmlift2.h	. . . 4 |- H = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( f ` 0 ) = P ) )
7		cvmlift2.k	. . . 4 |- K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) )
8		fveq2 6191	. . . . . 6 |- ( a = z -> ( ( ( II tX II ) CnP C ) ` a ) = ( ( ( II tX II ) CnP C ) ` z ) )
9	8	eleq2d 2687	. . . . 5 |- ( a = z -> ( K e. ( ( ( II tX II ) CnP C ) ` a ) <-> K e. ( ( ( II tX II ) CnP C ) ` z ) ) )
10	9	cbvrabv 3199	. . . 4 |- { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } = { z e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` z ) }
11		sneq 4187	. . . . . . 7 |- ( z = b -> { z } = { b } )
12	11	xpeq2d 5139	. . . . . 6 |- ( z = b -> ( ( 0 [,] 1 ) X. { z } ) = ( ( 0 [,] 1 ) X. { b } ) )
13	12	sseq1d 3632	. . . . 5 |- ( z = b -> ( ( ( 0 [,] 1 ) X. { z } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( ( 0 [,] 1 ) X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) )
14	13	cbvrabv 3199	. . . 4 |- { z e. ( 0 [,] 1 ) | ( ( 0 [,] 1 ) X. { z } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } } = { b e. ( 0 [,] 1 ) | ( ( 0 [,] 1 ) X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } }
15		simpr 477	. . . . . . 7 |- ( ( c = r /\ d = t ) -> d = t )
16	15	eleq1d 2686	. . . . . 6 |- ( ( c = r /\ d = t ) -> ( d e. ( 0 [,] 1 ) <-> t e. ( 0 [,] 1 ) ) )
17		xpeq1 5128	. . . . . . . . . 10 |- ( v = u -> ( v X. { b } ) = ( u X. { b } ) )
18	17	sseq1d 3632	. . . . . . . . 9 |- ( v = u -> ( ( v X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) )
19		xpeq1 5128	. . . . . . . . . 10 |- ( v = u -> ( v X. { d } ) = ( u X. { d } ) )
20	19	sseq1d 3632	. . . . . . . . 9 |- ( v = u -> ( ( v X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) )
21	18, 20	bibi12d 335	. . . . . . . 8 |- ( v = u -> ( ( ( v X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( v X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) <-> ( ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) ) )
22	21	cbvrexv 3172	. . . . . . 7 |- ( E. v e. ( ( nei ` II ) ` { c } ) ( ( v X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( v X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) <-> E. u e. ( ( nei ` II ) ` { c } ) ( ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) )
23		simpl 473	. . . . . . . . . 10 |- ( ( c = r /\ d = t ) -> c = r )
24	23	sneqd 4189	. . . . . . . . 9 |- ( ( c = r /\ d = t ) -> { c } = { r } )
25	24	fveq2d 6195	. . . . . . . 8 |- ( ( c = r /\ d = t ) -> ( ( nei ` II ) ` { c } ) = ( ( nei ` II ) ` { r } ) )
26	15	sneqd 4189	. . . . . . . . . . 11 |- ( ( c = r /\ d = t ) -> { d } = { t } )
27	26	xpeq2d 5139	. . . . . . . . . 10 |- ( ( c = r /\ d = t ) -> ( u X. { d } ) = ( u X. { t } ) )
28	27	sseq1d 3632	. . . . . . . . 9 |- ( ( c = r /\ d = t ) -> ( ( u X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { t } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) )
29	28	bibi2d 332	. . . . . . . 8 |- ( ( c = r /\ d = t ) -> ( ( ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) <-> ( ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { t } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) ) )
30	25, 29	rexeqbidv 3153	. . . . . . 7 |- ( ( c = r /\ d = t ) -> ( E. u e. ( ( nei ` II ) ` { c } ) ( ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) <-> E. u e. ( ( nei ` II ) ` { r } ) ( ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { t } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) ) )
31	22, 30	syl5bb 272	. . . . . 6 |- ( ( c = r /\ d = t ) -> ( E. v e. ( ( nei ` II ) ` { c } ) ( ( v X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( v X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) <-> E. u e. ( ( nei ` II ) ` { r } ) ( ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { t } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) ) )
32	16, 31	anbi12d 747	. . . . 5 |- ( ( c = r /\ d = t ) -> ( ( d e. ( 0 [,] 1 ) /\ E. v e. ( ( nei ` II ) ` { c } ) ( ( v X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( v X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) ) <-> ( t e. ( 0 [,] 1 ) /\ E. u e. ( ( nei ` II ) ` { r } ) ( ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { t } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) ) ) )
33	32	cbvopabv 4722	. . . 4 |- { <. c , d >. | ( d e. ( 0 [,] 1 ) /\ E. v e. ( ( nei ` II ) ` { c } ) ( ( v X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( v X. { d } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) ) } = { <. r , t >. | ( t e. ( 0 [,] 1 ) /\ E. u e. ( ( nei ` II ) ` { r } ) ( ( u X. { b } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } <-> ( u X. { t } ) C_ { a e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` a ) } ) ) }
34	1, 2, 3, 4, 5, 6, 7, 10, 14, 33	cvmlift2lem12 31296	. . 3 |- ( ph -> K e. ( ( II tX II ) Cn C ) )
35	1, 2, 3, 4, 5, 6, 7	cvmlift2lem7 31291	. . 3 |- ( ph -> ( F o. K ) = G )
36		0elunit 12290	. . . . 5 |- 0 e. ( 0 [,] 1 )
37	1, 2, 3, 4, 5, 6, 7	cvmlift2lem8 31292	. . . . 5 |- ( ( ph /\ 0 e. ( 0 [,] 1 ) ) -> ( 0 K 0 ) = ( H ` 0 ) )
38	36, 37	mpan2 707	. . . 4 |- ( ph -> ( 0 K 0 ) = ( H ` 0 ) )
39	1, 2, 3, 4, 5, 6	cvmlift2lem2 31286	. . . . 5 |- ( ph -> ( H e. ( II Cn C ) /\ ( F o. H ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( H ` 0 ) = P ) )
40	39	simp3d 1075	. . . 4 |- ( ph -> ( H ` 0 ) = P )
41	38, 40	eqtrd 2656	. . 3 |- ( ph -> ( 0 K 0 ) = P )
42		coeq2 5280	. . . . . 6 |- ( g = K -> ( F o. g ) = ( F o. K ) )
43	42	eqeq1d 2624	. . . . 5 |- ( g = K -> ( ( F o. g ) = G <-> ( F o. K ) = G ) )
44		oveq 6656	. . . . . 6 |- ( g = K -> ( 0 g 0 ) = ( 0 K 0 ) )
45	44	eqeq1d 2624	. . . . 5 |- ( g = K -> ( ( 0 g 0 ) = P <-> ( 0 K 0 ) = P ) )
46	43, 45	anbi12d 747	. . . 4 |- ( g = K -> ( ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) <-> ( ( F o. K ) = G /\ ( 0 K 0 ) = P ) ) )
47	46	rspcev 3309	. . 3 |- ( ( K e. ( ( II tX II ) Cn C ) /\ ( ( F o. K ) = G /\ ( 0 K 0 ) = P ) ) -> E. g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) )
48	34, 35, 41, 47	syl12anc 1324	. 2 |- ( ph -> E. g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) )
49		iitop 22683	. . . . 5 |- II e. Top
50		iiuni 22684	. . . . 5 |- ( 0 [,] 1 ) = U. II
51	49, 49, 50, 50	txunii 21396	. . . 4 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
52		iiconn 22690	. . . . . 6 |- II e. Conn
53		txconn 21492	. . . . . 6 |- ( ( II e. Conn /\ II e. Conn ) -> ( II tX II ) e. Conn )
54	52, 52, 53	mp2an 708	. . . . 5 |- ( II tX II ) e. Conn
55	54	a1i 11	. . . 4 |- ( ph -> ( II tX II ) e. Conn )
56		iinllyconn 31236	. . . . . 6 |- II e. 𝑛Locally Conn
57		txconn 21492	. . . . . . 7 |- ( ( x e. Conn /\ y e. Conn ) -> ( x tX y ) e. Conn )
58	57	txnlly 21440	. . . . . 6 |- ( ( II e. 𝑛Locally Conn /\ II e. 𝑛Locally Conn ) -> ( II tX II ) e. 𝑛Locally Conn )
59	56, 56, 58	mp2an 708	. . . . 5 |- ( II tX II ) e. 𝑛Locally Conn
60	59	a1i 11	. . . 4 |- ( ph -> ( II tX II ) e. 𝑛Locally Conn )
61		opelxpi 5148	. . . . . 6 |- ( ( 0 e. ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) ) -> <. 0 , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
62	36, 36, 61	mp2an 708	. . . . 5 |- <. 0 , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) )
63	62	a1i 11	. . . 4 |- ( ph -> <. 0 , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
64		df-ov 6653	. . . . 5 |- ( 0 G 0 ) = ( G ` <. 0 , 0 >. )
65	5, 64	syl6eq 2672	. . . 4 |- ( ph -> ( F ` P ) = ( G ` <. 0 , 0 >. ) )
66	1, 51, 2, 55, 60, 63, 3, 4, 65	cvmliftmo 31266	. . 3 |- ( ph -> E* g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( g ` <. 0 , 0 >. ) = P ) )
67		df-ov 6653	. . . . . 6 |- ( 0 g 0 ) = ( g ` <. 0 , 0 >. )
68	67	eqeq1i 2627	. . . . 5 |- ( ( 0 g 0 ) = P <-> ( g ` <. 0 , 0 >. ) = P )
69	68	anbi2i 730	. . . 4 |- ( ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) <-> ( ( F o. g ) = G /\ ( g ` <. 0 , 0 >. ) = P ) )
70	69	rmobii 3133	. . 3 |- ( E* g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) <-> E* g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( g ` <. 0 , 0 >. ) = P ) )
71	66, 70	sylibr 224	. 2 |- ( ph -> E* g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) )
72		reu5 3159	. 2 |- ( E! g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) <-> ( E. g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) /\ E* g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) ) )
73	48, 71, 72	sylanbrc 698	1 |- ( ph -> E! g e. ( ( II tX II ) Cn C ) ( ( F o. g ) = G /\ ( 0 g 0 ) = P ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   E!wreu 2914   E*wrmo 2915   {crab 2916   C_ wss 3574   {csn 4177   <.cop 4183   U.cuni 4436   {copab 4712   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   [,]cicc 12178   neicnei 20901   Cn ccn 21028   CnP ccnp 21029  Conncconn 21214  𝑛Locally cnlly 21268   tX ctx 21363   IIcii 22678   CovMap ccvm 31237

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  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-rmo 2920  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-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-cnp 21032  df-cmp 21190  df-conn 21215  df-lly 21269  df-nlly 21270  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791  df-pconn 31203  df-sconn 31204  df-cvm 31238

This theorem is referenced by:  cvmlift2  31298