Theorem restopnb 12350

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 989	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ B )
2		simpr2 988	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> B C_ A )
3	1, 2	sstrd 3107	. . . . . 6 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ A )
4		df-ss 3084	. . . . . 6 |- ( C C_ A <-> ( C i^i A ) = C )
5	3, 4	sylib 121	. . . . 5 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> ( C i^i A ) = C )
6	5	eqcomd 2145	. . . 4 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C = ( C i^i A ) )
7		ineq1 3270	. . . . . 6 |- ( v = C -> ( v i^i A ) = ( C i^i A ) )
8	7	rspceeqv 2807	. . . . 5 |- ( ( C e. J /\ C = ( C i^i A ) ) -> E. v e. J C = ( v i^i A ) )
9	8	expcom 115	. . . 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 3286	. . . . . 6 |- ( ( v i^i A ) i^i B ) = ( v i^i ( A i^i B ) )
12		simprr 521	. . . . . . . 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 3276	. . . . . . 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 1025	. . . . . . . . 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 3084	. . . . . . . . 9 |- ( C C_ B <-> ( C i^i B ) = C )
16	14, 15	sylib 121	. . . . . . . 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 470	. . . . . . 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 2174	. . . . . 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 1024	. . . . . . . . 9 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> B C_ A )
20		sseqin2 3295	. . . . . . . . 9 |- ( B C_ A <-> ( A i^i B ) = B )
21	19, 20	sylib 121	. . . . . . . 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 3277	. . . . . . 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 470	. . . . . 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 2196	. . . . 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 522	. . . . . 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 520	. . . . . 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 1023	. . . . . 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 12170	. . . . . 6 |- ( ( J e. Top /\ v e. J /\ B e. J ) -> ( v i^i B ) e. J )
29	25, 26, 27, 28	syl3anc 1216	. . . . 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 2216	. . . 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 2550	. . 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 128	. 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 12127	. . 3 |- ( ( J e. Top /\ A e. V ) -> ( C e. ( J ↾t A ) <-> E. v e. J C = ( v i^i A ) ) )
34	33	adantr 274	. 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 190	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 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2417   i^i cin 3070   C_ wss 3071  (class class class)co 5774   ↾_t crest 12120   Topctop 12164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-rest 12122  df-top 12165

This theorem is referenced by:  restopn2  12352