Theorem 2ndcsb 21252

Description: Having a countable subbase is a sufficient condition for second-countability. (Contributed by Jeff Hankins, 17-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
2ndcsb	|- ( J e. 2ndc <-> E. x ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) )

Distinct variable group:   x, J

Proof of Theorem 2ndcsb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		is2ndc 21249	. . 3 |- ( J e. 2ndc <-> E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = J ) )
2		df-rex 2918	. . . 4 |- ( E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = J ) <-> E. x ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) )
3		simprl 794	. . . . . 6 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> x ~<_ om )
4		ssfii 8325	. . . . . . . . 9 |- ( x e. TopBases -> x C_ ( fi ` x ) )
5	4	adantr 481	. . . . . . . 8 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> x C_ ( fi ` x ) )
6		fvex 6201	. . . . . . . . . 10 |- ( topGen ` x ) e. _V
7		bastg 20770	. . . . . . . . . . 11 |- ( x e. TopBases -> x C_ ( topGen ` x ) )
8	7	adantr 481	. . . . . . . . . 10 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> x C_ ( topGen ` x ) )
9		fiss 8330	. . . . . . . . . 10 |- ( ( ( topGen ` x ) e. _V /\ x C_ ( topGen ` x ) ) -> ( fi ` x ) C_ ( fi ` ( topGen ` x ) ) )
10	6, 8, 9	sylancr 695	. . . . . . . . 9 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> ( fi ` x ) C_ ( fi ` ( topGen ` x ) ) )
11		tgcl 20773	. . . . . . . . . . 11 |- ( x e. TopBases -> ( topGen ` x ) e. Top )
12	11	adantr 481	. . . . . . . . . 10 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> ( topGen ` x ) e. Top )
13		fitop 20705	. . . . . . . . . 10 |- ( ( topGen ` x ) e. Top -> ( fi ` ( topGen ` x ) ) = ( topGen ` x ) )
14	12, 13	syl 17	. . . . . . . . 9 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> ( fi ` ( topGen ` x ) ) = ( topGen ` x ) )
15	10, 14	sseqtrd 3641	. . . . . . . 8 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> ( fi ` x ) C_ ( topGen ` x ) )
16		2basgen 20794	. . . . . . . 8 |- ( ( x C_ ( fi ` x ) /\ ( fi ` x ) C_ ( topGen ` x ) ) -> ( topGen ` x ) = ( topGen ` ( fi ` x ) ) )
17	5, 15, 16	syl2anc 693	. . . . . . 7 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> ( topGen ` x ) = ( topGen ` ( fi ` x ) ) )
18		simprr 796	. . . . . . 7 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> ( topGen ` x ) = J )
19	17, 18	eqtr3d 2658	. . . . . 6 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> ( topGen ` ( fi ` x ) ) = J )
20	3, 19	jca 554	. . . . 5 |- ( ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) )
21	20	eximi 1762	. . . 4 |- ( E. x ( x e. TopBases /\ ( x ~<_ om /\ ( topGen ` x ) = J ) ) -> E. x ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) )
22	2, 21	sylbi 207	. . 3 |- ( E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = J ) -> E. x ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) )
23	1, 22	sylbi 207	. 2 |- ( J e. 2ndc -> E. x ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) )
24		fibas 20781	. . . . 5 |- ( fi ` x ) e. TopBases
25		simpl 473	. . . . . . 7 |- ( ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) -> x ~<_ om )
26		vex 3203	. . . . . . . 8 |- x e. _V
27		fictb 9067	. . . . . . . 8 |- ( x e. _V -> ( x ~<_ om <-> ( fi ` x ) ~<_ om ) )
28	26, 27	ax-mp 5	. . . . . . 7 |- ( x ~<_ om <-> ( fi ` x ) ~<_ om )
29	25, 28	sylib 208	. . . . . 6 |- ( ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) -> ( fi ` x ) ~<_ om )
30		simpr 477	. . . . . 6 |- ( ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) -> ( topGen ` ( fi ` x ) ) = J )
31	29, 30	jca 554	. . . . 5 |- ( ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) -> ( ( fi ` x ) ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) )
32		breq1 4656	. . . . . . 7 |- ( y = ( fi ` x ) -> ( y ~<_ om <-> ( fi ` x ) ~<_ om ) )
33		fveq2 6191	. . . . . . . 8 |- ( y = ( fi ` x ) -> ( topGen ` y ) = ( topGen ` ( fi ` x ) ) )
34	33	eqeq1d 2624	. . . . . . 7 |- ( y = ( fi ` x ) -> ( ( topGen ` y ) = J <-> ( topGen ` ( fi ` x ) ) = J ) )
35	32, 34	anbi12d 747	. . . . . 6 |- ( y = ( fi ` x ) -> ( ( y ~<_ om /\ ( topGen ` y ) = J ) <-> ( ( fi ` x ) ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) ) )
36	35	rspcev 3309	. . . . 5 |- ( ( ( fi ` x ) e. TopBases /\ ( ( fi ` x ) ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) ) -> E. y e. TopBases ( y ~<_ om /\ ( topGen ` y ) = J ) )
37	24, 31, 36	sylancr 695	. . . 4 |- ( ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) -> E. y e. TopBases ( y ~<_ om /\ ( topGen ` y ) = J ) )
38		is2ndc 21249	. . . 4 |- ( J e. 2ndc <-> E. y e. TopBases ( y ~<_ om /\ ( topGen ` y ) = J ) )
39	37, 38	sylibr 224	. . 3 |- ( ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) -> J e. 2ndc )
40	39	exlimiv 1858	. 2 |- ( E. x ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) -> J e. 2ndc )
41	23, 40	impbii 199	1 |- ( J e. 2ndc <-> E. x ( x ~<_ om /\ ( topGen ` ( fi ` x ) ) = J ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   ` cfv 5888   omcom 7065   ~<_ cdom 7953   ficfi 8316   topGenctg 16098   Topctop 20698   TopBasesctb 20749   2ndcc2ndc 21241

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

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-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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-card 8765  df-acn 8768  df-cda 8990  df-topgen 16104  df-top 20699  df-bases 20750  df-2ndc 21243

This theorem is referenced by: (None)