Theorem alexsubALTlem1 21851

Description: Lemma for alexsubALT 21855. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.)

Hypothesis

Ref	Expression
alexsubALT.1	|- X = U. J

Assertion

Ref	Expression
alexsubALTlem1	|- ( J e. Comp -> E. x ( J = ( topGen ` ( fi ` x ) ) /\ A. c e. ~P x ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) )

Distinct variable groups:   c, d, x, J   X, c, d, x

Proof of Theorem alexsubALTlem1

Step	Hyp	Ref	Expression
1		cmptop 21198	. . 3 |- ( J e. Comp -> J e. Top )
2		fitop 20705	. . . . 5 |- ( J e. Top -> ( fi ` J ) = J )
3	2	fveq2d 6195	. . . 4 |- ( J e. Top -> ( topGen ` ( fi ` J ) ) = ( topGen ` J ) )
4		tgtop 20777	. . . 4 |- ( J e. Top -> ( topGen ` J ) = J )
5	3, 4	eqtr2d 2657	. . 3 |- ( J e. Top -> J = ( topGen ` ( fi ` J ) ) )
6	1, 5	syl 17	. 2 |- ( J e. Comp -> J = ( topGen ` ( fi ` J ) ) )
7		selpw 4165	. . . 4 |- ( c e. ~P J <-> c C_ J )
8		alexsubALT.1	. . . . . 6 |- X = U. J
9	8	cmpcov 21192	. . . . 5 |- ( ( J e. Comp /\ c C_ J /\ X = U. c ) -> E. d e. ( ~P c i^i Fin ) X = U. d )
10	9	3exp 1264	. . . 4 |- ( J e. Comp -> ( c C_ J -> ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) )
11	7, 10	syl5bi 232	. . 3 |- ( J e. Comp -> ( c e. ~P J -> ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) )
12	11	ralrimiv 2965	. 2 |- ( J e. Comp -> A. c e. ~P J ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) )
13		fveq2 6191	. . . . . 6 |- ( x = J -> ( fi ` x ) = ( fi ` J ) )
14	13	fveq2d 6195	. . . . 5 |- ( x = J -> ( topGen ` ( fi ` x ) ) = ( topGen ` ( fi ` J ) ) )
15	14	eqeq2d 2632	. . . 4 |- ( x = J -> ( J = ( topGen ` ( fi ` x ) ) <-> J = ( topGen ` ( fi ` J ) ) ) )
16		pweq 4161	. . . . 5 |- ( x = J -> ~P x = ~P J )
17	16	raleqdv 3144	. . . 4 |- ( x = J -> ( A. c e. ~P x ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) <-> A. c e. ~P J ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) )
18	15, 17	anbi12d 747	. . 3 |- ( x = J -> ( ( J = ( topGen ` ( fi ` x ) ) /\ A. c e. ~P x ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) <-> ( J = ( topGen ` ( fi ` J ) ) /\ A. c e. ~P J ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) ) )
19	18	spcegv 3294	. 2 |- ( J e. Comp -> ( ( J = ( topGen ` ( fi ` J ) ) /\ A. c e. ~P J ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) -> E. x ( J = ( topGen ` ( fi ` x ) ) /\ A. c e. ~P x ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) ) )
20	6, 12, 19	mp2and 715	1 |- ( J e. Comp -> E. x ( J = ( topGen ` ( fi ` x ) ) /\ A. c e. ~P x ( X = U. c -> E. d e. ( ~P c i^i Fin ) X = U. d ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ` cfv 5888   Fincfn 7955   ficfi 8316   topGenctg 16098   Topctop 20698   Compccmp 21189

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-3or 1038  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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-topgen 16104  df-top 20699  df-cmp 21190

This theorem is referenced by:  alexsubALT  21855