Theorem restopnb 14855

Description: If B is an open subset of the subspace base set A, then any subset of B is open iff it is open in A. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
restopnb	|- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J <-> C e. ( J ↾t A ) ) )

Proof of Theorem restopnb

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr3 1029	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ B )
2		simpr2 1028	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> B C_ A )
3	1, 2	sstrd 3234	. . . . . 6 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ A )
4		df-ss 3210	. . . . . 6 |- ( C C_ A <-> ( C i^i A ) = C )
5	3, 4	sylib 122	. . . . 5 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C i^i A ) = C )
6	5	eqcomd 2235	. . . 4 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C = ( C i^i A ) )
7		ineq1 3398	. . . . . 6 |- ( v = C -> ( v i^i A ) = ( C i^i A ) )
8	7	rspceeqv 2925	. . . . 5 |- ( ( C e. J /\ C = ( C i^i A ) ) -> E. v e. J C = ( v i^i A ) )
9	8	expcom 116	. . . 4 |- ( C = ( C i^i A ) -> ( C e. J -> E. v e. J C = ( v i^i A ) ) )
10	6, 9	syl 14	. . 3 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J -> E. v e. J C = ( v i^i A ) ) )
11		inass 3414	. . . . . 6 |- ( ( v i^i A ) i^i B ) = ( v i^i ( A i^i B ) )
12		simprr 531	. . . . . . . 8 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> C = ( v i^i A ) )
13	12	ineq1d 3404	. . . . . . 7 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( C i^i B ) = ( ( v i^i A ) i^i B ) )
14		simplr3 1065	. . . . . . . . 9 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> C C_ B )
15		df-ss 3210	. . . . . . . . 9 |- ( C C_ B <-> ( C i^i B ) = C )
16	14, 15	sylib 122	. . . . . . . 8 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> ( C i^i B ) = C )
17	16	adantrr 479	. . . . . . 7 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( C i^i B ) = C )
18	13, 17	eqtr3d 2264	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( ( v i^i A ) i^i B ) = C )
19		simplr2 1064	. . . . . . . . 9 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> B C_ A )
20		sseqin2 3423	. . . . . . . . 9 |- ( B C_ A <-> ( A i^i B ) = B )
21	19, 20	sylib 122	. . . . . . . 8 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> ( A i^i B ) = B )
22	21	ineq2d 3405	. . . . . . 7 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> ( v i^i ( A i^i B ) ) = ( v i^i B ) )
23	22	adantrr 479	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( v i^i ( A i^i B ) ) = ( v i^i B ) )
24	11, 18, 23	3eqtr3a 2286	. . . . 5 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> C = ( v i^i B ) )
25		simplll 533	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> J e. Top )
26		simprl 529	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> v e. J )
27		simplr1 1063	. . . . . 6 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> B e. J )
28		inopn 14677	. . . . . 6 |- ( ( J e. Top /\ v e. J /\ B e. J ) -> ( v i^i B ) e. J )
29	25, 26, 27, 28	syl3anc 1271	. . . . 5 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> ( v i^i B ) e. J )
30	24, 29	eqeltrd 2306	. . . 4 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ ( v e. J /\ C = ( v i^i A ) ) ) -> C e. J )
31	30	rexlimdvaa 2649	. . 3 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( E. v e. J C = ( v i^i A ) -> C e. J ) )
32	10, 31	impbid 129	. 2 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J <-> E. v e. J C = ( v i^i A ) ) )
33		elrest 13279	. . 3 |- ( ( J e. Top /\ A e. V ) -> ( C e. ( J ↾t A ) <-> E. v e. J C = ( v i^i A ) ) )
34	33	adantr 276	. 2 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. ( J ↾t A ) <-> E. v e. J C = ( v i^i A ) ) )
35	32, 34	bitr4d 191	1 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C e. J <-> C e. ( J ↾t A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   i^i cin 3196   C_ wss 3197  (class class class)co 6001   ↾_t crest 13272   Topctop 14671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-rest 13274  df-top 14672

This theorem is referenced by:  restopn2  14857