Theorem cvmseu 31258

Description: Every element in U. T is a member of a unique element of T. (Contributed by Mario Carneiro, 14-Feb-2015.)

Hypotheses

Ref	Expression
cvmcov.1	|- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )
cvmseu.1	|- B = U. C

Assertion

Ref	Expression
cvmseu	|- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> E! x e. T A e. x )

Distinct variable groups:   k, s, u, v, x, C   k, F, s, u, v, x   k, J, s, u, v, x   x, S   U, k, s, u, v, x   T, s, u, v, x   u, A, v, x   v, B, x

Allowed substitution hints:   A( k, s)   B( u, k, s)   S( v, u, k, s)   T( k)

Proof of Theorem cvmseu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr2 1068	. . . . 5 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> A e. B )
2		simpr3 1069	. . . . 5 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> ( F ` A ) e. U )
3		cvmcn 31244	. . . . . . 7 |- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) )
4	3	adantr 481	. . . . . 6 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> F e. ( C Cn J ) )
5		cvmseu.1	. . . . . . 7 |- B = U. C
6		eqid 2622	. . . . . . 7 |- U. J = U. J
7	5, 6	cnf 21050	. . . . . 6 |- ( F e. ( C Cn J ) -> F : B --> U. J )
8		ffn 6045	. . . . . 6 |- ( F : B --> U. J -> F Fn B )
9		elpreima 6337	. . . . . 6 |- ( F Fn B -> ( A e. ( `' F " U ) <-> ( A e. B /\ ( F ` A ) e. U ) ) )
10	4, 7, 8, 9	4syl 19	. . . . 5 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> ( A e. ( `' F " U ) <-> ( A e. B /\ ( F ` A ) e. U ) ) )
11	1, 2, 10	mpbir2and 957	. . . 4 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> A e. ( `' F " U ) )
12		simpr1 1067	. . . . 5 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> T e. ( S ` U ) )
13		cvmcov.1	. . . . . 6 |- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )
14	13	cvmsuni 31251	. . . . 5 |- ( T e. ( S ` U ) -> U. T = ( `' F " U ) )
15	12, 14	syl 17	. . . 4 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> U. T = ( `' F " U ) )
16	11, 15	eleqtrrd 2704	. . 3 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> A e. U. T )
17		eluni2 4440	. . 3 |- ( A e. U. T <-> E. x e. T A e. x )
18	16, 17	sylib 208	. 2 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> E. x e. T A e. x )
19		inelcm 4032	. . . 4 |- ( ( A e. x /\ A e. z ) -> ( x i^i z ) =/= (/) )
20	13	cvmsdisj 31252	. . . . . . . 8 |- ( ( T e. ( S ` U ) /\ x e. T /\ z e. T ) -> ( x = z \/ ( x i^i z ) = (/) ) )
21	20	3expb 1266	. . . . . . 7 |- ( ( T e. ( S ` U ) /\ ( x e. T /\ z e. T ) ) -> ( x = z \/ ( x i^i z ) = (/) ) )
22	12, 21	sylan 488	. . . . . 6 |- ( ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) /\ ( x e. T /\ z e. T ) ) -> ( x = z \/ ( x i^i z ) = (/) ) )
23	22	ord 392	. . . . 5 |- ( ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) /\ ( x e. T /\ z e. T ) ) -> ( -. x = z -> ( x i^i z ) = (/) ) )
24	23	necon1ad 2811	. . . 4 |- ( ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) /\ ( x e. T /\ z e. T ) ) -> ( ( x i^i z ) =/= (/) -> x = z ) )
25	19, 24	syl5 34	. . 3 |- ( ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) /\ ( x e. T /\ z e. T ) ) -> ( ( A e. x /\ A e. z ) -> x = z ) )
26	25	ralrimivva 2971	. 2 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> A. x e. T A. z e. T ( ( A e. x /\ A e. z ) -> x = z ) )
27		eleq2 2690	. . 3 |- ( x = z -> ( A e. x <-> A e. z ) )
28	27	reu4 3400	. 2 |- ( E! x e. T A e. x <-> ( E. x e. T A e. x /\ A. x e. T A. z e. T ( ( A e. x /\ A e. z ) -> x = z ) ) )
29	18, 26, 28	sylanbrc 698	1 |- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` U ) /\ A e. B /\ ( F ` A ) e. U ) ) -> E! x e. T A e. x )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   \ cdif 3571   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Cn ccn 21028   Homeochmeo 21556   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-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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-map 7859  df-top 20699  df-topon 20716  df-cn 21031  df-cvm 31238

This theorem is referenced by:  cvmsiota  31259