Theorem cvmlift2lem11 31295

Description: Lemma for cvmlift2 31298. (Contributed by Mario Carneiro, 1-Jun-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 ) )
cvmlift2.m	|- M = { z e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` z ) }
cvmlift2lem11.1	|- ( ph -> U e. II )
cvmlift2lem11.2	|- ( ph -> V e. II )
cvmlift2lem11.3	|- ( ph -> Y e. V )
cvmlift2lem11.4	|- ( ph -> Z e. V )
cvmlift2lem11.5	|- ( ph -> ( E. w e. V ( K |` ( U X. { w } ) ) e. ( ( ( II tX II ) ↾t ( U X. { w } ) ) Cn C ) -> ( K |` ( U X. V ) ) e. ( ( ( II tX II ) ↾t ( U X. V ) ) Cn C ) ) )

Assertion

Ref	Expression
cvmlift2lem11	|- ( ph -> ( ( U X. { Y } ) C_ M -> ( U X. { Z } ) C_ M ) )

Distinct variable groups:   w, f, x, y, z, F   ph, f, w, x, y, z   x, M, y, z   f, J, w, x, y, z   w, U, z   f, G, w, x, y, z   w, V   f, H, w, x, y, z   z, Z   C, f, w, x, y, z   P, f, x, y, z   w, B, x, y, z   f, Y, w, x, y, z   f, K, w, x, y, z

Allowed substitution hints:   B( f)   P( w)   U( x, y, f)   M( w, f)   V( x, y, z, f)   Z( x, y, w, f)

Proof of Theorem cvmlift2lem11

Step	Hyp	Ref	Expression
1		cvmlift2lem11.1	. . . . . . 7 |- ( ph -> U e. II )
2	1	adantr 481	. . . . . 6 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> U e. II )
3		elssuni 4467	. . . . . . 7 |- ( U e. II -> U C_ U. II )
4		iiuni 22684	. . . . . . 7 |- ( 0 [,] 1 ) = U. II
5	3, 4	syl6sseqr 3652	. . . . . 6 |- ( U e. II -> U C_ ( 0 [,] 1 ) )
6	2, 5	syl 17	. . . . 5 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> U C_ ( 0 [,] 1 ) )
7		cvmlift2lem11.4	. . . . . . . 8 |- ( ph -> Z e. V )
8		cvmlift2lem11.2	. . . . . . . 8 |- ( ph -> V e. II )
9		elunii 4441	. . . . . . . . 9 |- ( ( Z e. V /\ V e. II ) -> Z e. U. II )
10	9, 4	syl6eleqr 2712	. . . . . . . 8 |- ( ( Z e. V /\ V e. II ) -> Z e. ( 0 [,] 1 ) )
11	7, 8, 10	syl2anc 693	. . . . . . 7 |- ( ph -> Z e. ( 0 [,] 1 ) )
12	11	adantr 481	. . . . . 6 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> Z e. ( 0 [,] 1 ) )
13	12	snssd 4340	. . . . 5 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> { Z } C_ ( 0 [,] 1 ) )
14		xpss12 5225	. . . . 5 |- ( ( U C_ ( 0 [,] 1 ) /\ { Z } C_ ( 0 [,] 1 ) ) -> ( U X. { Z } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
15	6, 13, 14	syl2anc 693	. . . 4 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. { Z } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
16		cvmlift2lem11.3	. . . . . . . . . 10 |- ( ph -> Y e. V )
17	16	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> Y e. V )
18		cvmlift2.b	. . . . . . . . . . . . 13 |- B = U. C
19		cvmlift2.f	. . . . . . . . . . . . 13 |- ( ph -> F e. ( C CovMap J ) )
20		cvmlift2.g	. . . . . . . . . . . . 13 |- ( ph -> G e. ( ( II tX II ) Cn J ) )
21		cvmlift2.p	. . . . . . . . . . . . 13 |- ( ph -> P e. B )
22		cvmlift2.i	. . . . . . . . . . . . 13 |- ( ph -> ( F ` P ) = ( 0 G 0 ) )
23		cvmlift2.h	. . . . . . . . . . . . 13 |- H = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( f ` 0 ) = P ) )
24		cvmlift2.k	. . . . . . . . . . . . 13 |- 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 ) )
25	18, 19, 20, 21, 22, 23, 24	cvmlift2lem5 31289	. . . . . . . . . . . 12 |- ( ph -> K : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B )
26	25	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> K : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B )
27	8	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> V e. II )
28		elssuni 4467	. . . . . . . . . . . . . . . 16 |- ( V e. II -> V C_ U. II )
29	28, 4	syl6sseqr 3652	. . . . . . . . . . . . . . 15 |- ( V e. II -> V C_ ( 0 [,] 1 ) )
30	27, 29	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> V C_ ( 0 [,] 1 ) )
31	30, 17	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> Y e. ( 0 [,] 1 ) )
32	31	snssd 4340	. . . . . . . . . . . 12 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> { Y } C_ ( 0 [,] 1 ) )
33		xpss12 5225	. . . . . . . . . . . 12 |- ( ( U C_ ( 0 [,] 1 ) /\ { Y } C_ ( 0 [,] 1 ) ) -> ( U X. { Y } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
34	6, 32, 33	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. { Y } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
35	26, 34	fssresd 6071	. . . . . . . . . 10 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( K |` ( U X. { Y } ) ) : ( U X. { Y } ) --> B )
36	34	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Y } ) ) -> ( U X. { Y } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
37		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Y } ) ) -> z e. ( U X. { Y } ) )
38		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. { Y } ) C_ M )
39		cvmlift2.m	. . . . . . . . . . . . . . 15 |- M = { z e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` z ) }
40	38, 39	syl6sseq 3651	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. { Y } ) C_ { z e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` z ) } )
41		ssrab 3680	. . . . . . . . . . . . . . 15 |- ( ( U X. { Y } ) C_ { z e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` z ) } <-> ( ( U X. { Y } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) /\ A. z e. ( U X. { Y } ) K e. ( ( ( II tX II ) CnP C ) ` z ) ) )
42	41	simprbi 480	. . . . . . . . . . . . . 14 |- ( ( U X. { Y } ) C_ { z e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` z ) } -> A. z e. ( U X. { Y } ) K e. ( ( ( II tX II ) CnP C ) ` z ) )
43	40, 42	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> A. z e. ( U X. { Y } ) K e. ( ( ( II tX II ) CnP C ) ` z ) )
44	43	r19.21bi 2932	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Y } ) ) -> K e. ( ( ( II tX II ) CnP C ) ` z ) )
45		iitopon 22682	. . . . . . . . . . . . . . 15 |- II e. (TopOn ` ( 0 [,] 1 ) )
46		txtopon 21394	. . . . . . . . . . . . . . 15 |- ( ( II e. (TopOn ` ( 0 [,] 1 ) ) /\ II e. (TopOn ` ( 0 [,] 1 ) ) ) -> ( II tX II ) e. (TopOn ` ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) )
47	45, 45, 46	mp2an 708	. . . . . . . . . . . . . 14 |- ( II tX II ) e. (TopOn ` ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
48	47	toponunii 20721	. . . . . . . . . . . . 13 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
49	48	cnpresti 21092	. . . . . . . . . . . 12 |- ( ( ( U X. { Y } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) /\ z e. ( U X. { Y } ) /\ K e. ( ( ( II tX II ) CnP C ) ` z ) ) -> ( K |` ( U X. { Y } ) ) e. ( ( ( ( II tX II ) ↾t ( U X. { Y } ) ) CnP C ) ` z ) )
50	36, 37, 44, 49	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Y } ) ) -> ( K |` ( U X. { Y } ) ) e. ( ( ( ( II tX II ) ↾t ( U X. { Y } ) ) CnP C ) ` z ) )
51	50	ralrimiva 2966	. . . . . . . . . 10 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> A. z e. ( U X. { Y } ) ( K |` ( U X. { Y } ) ) e. ( ( ( ( II tX II ) ↾t ( U X. { Y } ) ) CnP C ) ` z ) )
52		resttopon 20965	. . . . . . . . . . . 12 |- ( ( ( II tX II ) e. (TopOn ` ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) /\ ( U X. { Y } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( II tX II ) ↾t ( U X. { Y } ) ) e. (TopOn ` ( U X. { Y } ) ) )
53	47, 34, 52	sylancr 695	. . . . . . . . . . 11 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( ( II tX II ) ↾t ( U X. { Y } ) ) e. (TopOn ` ( U X. { Y } ) ) )
54		cvmtop1 31242	. . . . . . . . . . . . . 14 |- ( F e. ( C CovMap J ) -> C e. Top )
55	19, 54	syl 17	. . . . . . . . . . . . 13 |- ( ph -> C e. Top )
56	55	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> C e. Top )
57	18	toptopon 20722	. . . . . . . . . . . 12 |- ( C e. Top <-> C e. (TopOn ` B ) )
58	56, 57	sylib 208	. . . . . . . . . . 11 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> C e. (TopOn ` B ) )
59		cncnp 21084	. . . . . . . . . . 11 |- ( ( ( ( II tX II ) ↾t ( U X. { Y } ) ) e. (TopOn ` ( U X. { Y } ) ) /\ C e. (TopOn ` B ) ) -> ( ( K |` ( U X. { Y } ) ) e. ( ( ( II tX II ) ↾t ( U X. { Y } ) ) Cn C ) <-> ( ( K |` ( U X. { Y } ) ) : ( U X. { Y } ) --> B /\ A. z e. ( U X. { Y } ) ( K |` ( U X. { Y } ) ) e. ( ( ( ( II tX II ) ↾t ( U X. { Y } ) ) CnP C ) ` z ) ) ) )
60	53, 58, 59	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( ( K |` ( U X. { Y } ) ) e. ( ( ( II tX II ) ↾t ( U X. { Y } ) ) Cn C ) <-> ( ( K |` ( U X. { Y } ) ) : ( U X. { Y } ) --> B /\ A. z e. ( U X. { Y } ) ( K |` ( U X. { Y } ) ) e. ( ( ( ( II tX II ) ↾t ( U X. { Y } ) ) CnP C ) ` z ) ) ) )
61	35, 51, 60	mpbir2and 957	. . . . . . . . 9 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( K |` ( U X. { Y } ) ) e. ( ( ( II tX II ) ↾t ( U X. { Y } ) ) Cn C ) )
62		sneq 4187	. . . . . . . . . . . . 13 |- ( w = Y -> { w } = { Y } )
63	62	xpeq2d 5139	. . . . . . . . . . . 12 |- ( w = Y -> ( U X. { w } ) = ( U X. { Y } ) )
64	63	reseq2d 5396	. . . . . . . . . . 11 |- ( w = Y -> ( K |` ( U X. { w } ) ) = ( K |` ( U X. { Y } ) ) )
65	63	oveq2d 6666	. . . . . . . . . . . 12 |- ( w = Y -> ( ( II tX II ) ↾t ( U X. { w } ) ) = ( ( II tX II ) ↾t ( U X. { Y } ) ) )
66	65	oveq1d 6665	. . . . . . . . . . 11 |- ( w = Y -> ( ( ( II tX II ) ↾t ( U X. { w } ) ) Cn C ) = ( ( ( II tX II ) ↾t ( U X. { Y } ) ) Cn C ) )
67	64, 66	eleq12d 2695	. . . . . . . . . 10 |- ( w = Y -> ( ( K |` ( U X. { w } ) ) e. ( ( ( II tX II ) ↾t ( U X. { w } ) ) Cn C ) <-> ( K |` ( U X. { Y } ) ) e. ( ( ( II tX II ) ↾t ( U X. { Y } ) ) Cn C ) ) )
68	67	rspcev 3309	. . . . . . . . 9 |- ( ( Y e. V /\ ( K |` ( U X. { Y } ) ) e. ( ( ( II tX II ) ↾t ( U X. { Y } ) ) Cn C ) ) -> E. w e. V ( K |` ( U X. { w } ) ) e. ( ( ( II tX II ) ↾t ( U X. { w } ) ) Cn C ) )
69	17, 61, 68	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> E. w e. V ( K |` ( U X. { w } ) ) e. ( ( ( II tX II ) ↾t ( U X. { w } ) ) Cn C ) )
70		cvmlift2lem11.5	. . . . . . . . 9 |- ( ph -> ( E. w e. V ( K |` ( U X. { w } ) ) e. ( ( ( II tX II ) ↾t ( U X. { w } ) ) Cn C ) -> ( K |` ( U X. V ) ) e. ( ( ( II tX II ) ↾t ( U X. V ) ) Cn C ) ) )
71	70	imp 445	. . . . . . . 8 |- ( ( ph /\ E. w e. V ( K |` ( U X. { w } ) ) e. ( ( ( II tX II ) ↾t ( U X. { w } ) ) Cn C ) ) -> ( K |` ( U X. V ) ) e. ( ( ( II tX II ) ↾t ( U X. V ) ) Cn C ) )
72	69, 71	syldan 487	. . . . . . 7 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( K |` ( U X. V ) ) e. ( ( ( II tX II ) ↾t ( U X. V ) ) Cn C ) )
73	72	adantr 481	. . . . . 6 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Z } ) ) -> ( K |` ( U X. V ) ) e. ( ( ( II tX II ) ↾t ( U X. V ) ) Cn C ) )
74	7	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> Z e. V )
75	74	snssd 4340	. . . . . . . . 9 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> { Z } C_ V )
76		xpss2 5229	. . . . . . . . 9 |- ( { Z } C_ V -> ( U X. { Z } ) C_ ( U X. V ) )
77	75, 76	syl 17	. . . . . . . 8 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. { Z } ) C_ ( U X. V ) )
78		iitop 22683	. . . . . . . . . 10 |- II e. Top
79	78, 78	txtopi 21393	. . . . . . . . 9 |- ( II tX II ) e. Top
80		xpss12 5225	. . . . . . . . . 10 |- ( ( U C_ ( 0 [,] 1 ) /\ V C_ ( 0 [,] 1 ) ) -> ( U X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
81	6, 30, 80	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
82	48	restuni 20966	. . . . . . . . 9 |- ( ( ( II tX II ) e. Top /\ ( U X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( U X. V ) = U. ( ( II tX II ) ↾t ( U X. V ) ) )
83	79, 81, 82	sylancr 695	. . . . . . . 8 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. V ) = U. ( ( II tX II ) ↾t ( U X. V ) ) )
84	77, 83	sseqtrd 3641	. . . . . . 7 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. { Z } ) C_ U. ( ( II tX II ) ↾t ( U X. V ) ) )
85	84	sselda 3603	. . . . . 6 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Z } ) ) -> z e. U. ( ( II tX II ) ↾t ( U X. V ) ) )
86		eqid 2622	. . . . . . 7 |- U. ( ( II tX II ) ↾t ( U X. V ) ) = U. ( ( II tX II ) ↾t ( U X. V ) )
87	86	cncnpi 21082	. . . . . 6 |- ( ( ( K |` ( U X. V ) ) e. ( ( ( II tX II ) ↾t ( U X. V ) ) Cn C ) /\ z e. U. ( ( II tX II ) ↾t ( U X. V ) ) ) -> ( K |` ( U X. V ) ) e. ( ( ( ( II tX II ) ↾t ( U X. V ) ) CnP C ) ` z ) )
88	73, 85, 87	syl2anc 693	. . . . 5 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Z } ) ) -> ( K |` ( U X. V ) ) e. ( ( ( ( II tX II ) ↾t ( U X. V ) ) CnP C ) ` z ) )
89	79	a1i 11	. . . . . 6 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Z } ) ) -> ( II tX II ) e. Top )
90	81	adantr 481	. . . . . 6 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Z } ) ) -> ( U X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
91	78	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> II e. Top )
92		txopn 21405	. . . . . . . . . 10 |- ( ( ( II e. Top /\ II e. Top ) /\ ( U e. II /\ V e. II ) ) -> ( U X. V ) e. ( II tX II ) )
93	91, 91, 2, 27, 92	syl22anc 1327	. . . . . . . . 9 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. V ) e. ( II tX II ) )
94		isopn3i 20886	. . . . . . . . 9 |- ( ( ( II tX II ) e. Top /\ ( U X. V ) e. ( II tX II ) ) -> ( ( int ` ( II tX II ) ) ` ( U X. V ) ) = ( U X. V ) )
95	79, 93, 94	sylancr 695	. . . . . . . 8 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( ( int ` ( II tX II ) ) ` ( U X. V ) ) = ( U X. V ) )
96	77, 95	sseqtr4d 3642	. . . . . . 7 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. { Z } ) C_ ( ( int ` ( II tX II ) ) ` ( U X. V ) ) )
97	96	sselda 3603	. . . . . 6 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Z } ) ) -> z e. ( ( int ` ( II tX II ) ) ` ( U X. V ) ) )
98	25	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Z } ) ) -> K : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B )
99	48, 18	cnprest 21093	. . . . . 6 |- ( ( ( ( II tX II ) e. Top /\ ( U X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) /\ ( z e. ( ( int ` ( II tX II ) ) ` ( U X. V ) ) /\ K : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B ) ) -> ( K e. ( ( ( II tX II ) CnP C ) ` z ) <-> ( K |` ( U X. V ) ) e. ( ( ( ( II tX II ) ↾t ( U X. V ) ) CnP C ) ` z ) ) )
100	89, 90, 97, 98, 99	syl22anc 1327	. . . . 5 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Z } ) ) -> ( K e. ( ( ( II tX II ) CnP C ) ` z ) <-> ( K |` ( U X. V ) ) e. ( ( ( ( II tX II ) ↾t ( U X. V ) ) CnP C ) ` z ) ) )
101	88, 100	mpbird 247	. . . 4 |- ( ( ( ph /\ ( U X. { Y } ) C_ M ) /\ z e. ( U X. { Z } ) ) -> K e. ( ( ( II tX II ) CnP C ) ` z ) )
102	15, 101	ssrabdv 3681	. . 3 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. { Z } ) C_ { z e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) | K e. ( ( ( II tX II ) CnP C ) ` z ) } )
103	102, 39	syl6sseqr 3652	. 2 |- ( ( ph /\ ( U X. { Y } ) C_ M ) -> ( U X. { Z } ) C_ M )
104	103	ex 450	1 |- ( ph -> ( ( U X. { Y } ) C_ M -> ( U X. { Z } ) C_ M ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   {csn 4177   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   |` cres 5116   o. ccom 5118   -->wf 5884   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   [,]cicc 12178   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   intcnt 20821   Cn ccn 21028   CnP ccnp 21029   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:  cvmlift2lem12  31296