Theorem restopnb 14360

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 1007	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ B )
2		simpr2 1006	. . . . . . 7 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> B C_ A )
3	1, 2	sstrd 3190	. . . . . 6 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C C_ A )
4		df-ss 3167	. . . . . 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 2199	. . . 4 |- ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) -> C = ( C i^i A ) )
7		ineq1 3354	. . . . . 6 |- ( v = C -> ( v i^i A ) = ( C i^i A ) )
8	7	rspceeqv 2883	. . . . 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 3370	. . . . . 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 3360	. . . . . . 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 1043	. . . . . . . . 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 3167	. . . . . . . . 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 2228	. . . . . 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 1042	. . . . . . . . 9 |- ( ( ( ( J e. Top /\ A e. V ) /\ ( B e. J /\ B C_ A /\ C C_ B ) ) /\ v e. J ) -> B C_ A )
20		sseqin2 3379	. . . . . . . . 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 3361	. . . . . . 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 2250	. . . . 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 1041	. . . . . 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 14182	. . . . . 6 |- ( ( J e. Top /\ v e. J /\ B e. J ) -> ( v i^i B ) e. J )
29	25, 26, 27, 28	syl3anc 1249	. . . . 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 2270	. . . 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 2612	. . 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 12860	. . 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 980   = wceq 1364   e. wcel 2164   E.wrex 2473   i^i cin 3153   C_ wss 3154  (class class class)co 5919   ↾_t crest 12853   Topctop 14176

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-rest 12855  df-top 14177

This theorem is referenced by:  restopn2  14362