Theorem 2ndcrest 21257

Description: A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
2ndcrest	|- ( ( J e. 2ndc /\ A e. V ) -> ( J ↾t A ) e. 2ndc )

Proof of Theorem 2ndcrest

Dummy variables x y are mutually distinct and 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		simplr 792	. . . . . . . 8 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> x e. TopBases )
3		simpll 790	. . . . . . . 8 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> A e. V )
4		tgrest 20963	. . . . . . . 8 |- ( ( x e. TopBases /\ A e. V ) -> ( topGen ` ( x ↾t A ) ) = ( ( topGen ` x ) ↾t A ) )
5	2, 3, 4	syl2anc 693	. . . . . . 7 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> ( topGen ` ( x ↾t A ) ) = ( ( topGen ` x ) ↾t A ) )
6		restbas 20962	. . . . . . . . 9 |- ( x e. TopBases -> ( x ↾t A ) e. TopBases )
7	6	ad2antlr 763	. . . . . . . 8 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> ( x ↾t A ) e. TopBases )
8		restval 16087	. . . . . . . . . 10 |- ( ( x e. TopBases /\ A e. V ) -> ( x ↾t A ) = ran ( y e. x |-> ( y i^i A ) ) )
9	2, 3, 8	syl2anc 693	. . . . . . . . 9 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> ( x ↾t A ) = ran ( y e. x |-> ( y i^i A ) ) )
10		1stcrestlem 21255	. . . . . . . . . 10 |- ( x ~<_ om -> ran ( y e. x |-> ( y i^i A ) ) ~<_ om )
11	10	adantl 482	. . . . . . . . 9 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> ran ( y e. x |-> ( y i^i A ) ) ~<_ om )
12	9, 11	eqbrtrd 4675	. . . . . . . 8 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> ( x ↾t A ) ~<_ om )
13		2ndci 21251	. . . . . . . 8 |- ( ( ( x ↾t A ) e. TopBases /\ ( x ↾t A ) ~<_ om ) -> ( topGen ` ( x ↾t A ) ) e. 2ndc )
14	7, 12, 13	syl2anc 693	. . . . . . 7 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> ( topGen ` ( x ↾t A ) ) e. 2ndc )
15	5, 14	eqeltrrd 2702	. . . . . 6 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> ( ( topGen ` x ) ↾t A ) e. 2ndc )
16		oveq1 6657	. . . . . . 7 |- ( ( topGen ` x ) = J -> ( ( topGen ` x ) ↾t A ) = ( J ↾t A ) )
17	16	eleq1d 2686	. . . . . 6 |- ( ( topGen ` x ) = J -> ( ( ( topGen ` x ) ↾t A ) e. 2ndc <-> ( J ↾t A ) e. 2ndc ) )
18	15, 17	syl5ibcom 235	. . . . 5 |- ( ( ( A e. V /\ x e. TopBases ) /\ x ~<_ om ) -> ( ( topGen ` x ) = J -> ( J ↾t A ) e. 2ndc ) )
19	18	expimpd 629	. . . 4 |- ( ( A e. V /\ x e. TopBases ) -> ( ( x ~<_ om /\ ( topGen ` x ) = J ) -> ( J ↾t A ) e. 2ndc ) )
20	19	rexlimdva 3031	. . 3 |- ( A e. V -> ( E. x e. TopBases ( x ~<_ om /\ ( topGen ` x ) = J ) -> ( J ↾t A ) e. 2ndc ) )
21	1, 20	syl5bi 232	. 2 |- ( A e. V -> ( J e. 2ndc -> ( J ↾t A ) e. 2ndc ) )
22	21	impcom 446	1 |- ( ( J e. 2ndc /\ A e. V ) -> ( J ↾t A ) e. 2ndc )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   ↾_t crest 16081   topGenctg 16098   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-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-card 8765  df-acn 8768  df-rest 16083  df-topgen 16104  df-bases 20750  df-2ndc 21243

This theorem is referenced by: (None)